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few Approaches to Numeracy 

Lynn Arthur Steen, Editor 


On the Shoulders of 


New Approaches to Numeracy 


Mathematical Sciences Education Board 
National Research Council 

Washington, D.C. 1990 

NATIONAL ACADEMY PRESS 2101 Constitution Avenue, NW Washington, DC 20418 

NOTICE: The project that is the subject of this report was approved by the Governing Board of the 
National Research Council, whose members are drawn from the councils of the National Academy 
of Sciences, the National Academy of Engineering, and the Institute of Medicine. The authors of the 
report were chosen for their special competences. 

The National Research Council was organized by the National Academy of Sciences in 1916 to 
associate the broad community of science and technology with the Academy's purposes of furthering 
knowledge and advising the federal government. Functioning in accordance with general policies 
determined by the Academy, the Council has become the principal operating agency of both the 
National Academy of Sciences and the National Academy of Engineering in providing services to 
the government, the public, and the scientific and engineering communities. The Council is 
administered jointly by both Academies and the Institute of Medicine. Dr. Frank Press and Dr. 
Robert M. White are chairman and vice chairman, respectively, of the National Research Council. 

The Mathematical Sciences Education Board was established in 1985 to provide a continuing 
national overview and assessment capability for mathematics education and is concerned with 
mathematical sciences education for all students at all levels. The Board reports directly to the 
Governing Board of the National Research Council. 

Development, publication, and dissemination of this book were supported by a grant from the 
Carnegie Corporation of New York. Additional dissemination of the book was supported by a grant 
from the Andrew W. Mellon Foundation. The observations made herein do not necessarily reflect 
the views of the grantors. 

Library of Congress Cataloguing-in-Publication Data 

On the shoulders of giants : new approaches to numeracy / Lynn Arthur 
Steen, editor ; Mathematical Sciences Education Board, National 
Research Council. 

p. cm. 

Includes bibliographical references and index. 
ISBN 0-309-04234-8 (hardbound) 

1. Mathematics Study and teaching United States. I. Steen, 
Lynn Arthur, 1941- II. National Research Council (U.S.). 

Mathematical Sciences Education Board. 
QA13.053 1990 

513'.071073-^dc20 90-41566 

Copyright 1990 by the National Academy of Sciences 

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symmetrical pieces. Created by Thomas Banchoff and Nicholas Thompson at Brown University. 

Permission for limited reproduction of portions of this book for educational purposes but not for 
sale may be granted on receipt of a written request to the National Academy Press, 2101 Constitution 
Avenue, NW, Washington, DC 20418. 

Copies of this report may be purchased from the National Academy Press, 2101 Constitution 
Avenue, NW, Washington, DC 20418. 

Printed in the United States of America 


Today's headlines are filled with reports of illiteracy, innumeracy, and 
other signs of educational decay. Tomorrow's schools can be filled with 
evidence of renaissance if we begin now to till the soil for effective 
education in mathematics, in science, and in all subjects. This vol- 
ume offers five visions of mathematics suitable for tomorrow's schools 
visions rooted in imagination, in mathematics, and in science. Ideas in 
this volume can provide fertile soil for new approaches to tomorrow's 

Forces created by computers, applications, demographics, and schools 
themselves are changing profoundly the way mathematics is practiced, 
the way it is taught, and the way it is learned. Even as we work to make 
incremental change in today's schools, we must think also about more 
significant change that will be possible, indeed inevitable, in the future. 
For this reason the Mathematical Sciences Education Board (MSEB) 
decided that one of today's priorities is to stimulate imaginative thinking 
about tomorrow's curriculum. 

In this volume readers will find a vision of the richness of mathe- 
matics expressed through five vignettes that illustrate different possible 
strands of school mathematics. These papers expand on the theme of 
mathematics as the language and science of patterns and are introduced 
by a brief essay that highlights interconnections and common ideas. The 
authors were asked to explore ideas with deep roots in the mathematical 
sciences without concern for limitations of present schools or curricula. 
They do, however, suggest through numerous imaginative examples how 


mathematical ideas can be developed from informal childhood explo- 
ration through formal school and college study. 

The papers in this volume are intended as a vehicle to stimulate cre- 
ative approaches to mathematics curricula in the next century. The 
volume itself is part of a national dialogue on mathematics education 
stimulated by a series of recent publications: 

Everybody Counts: A Report to the Nation on the Future of 

Mathematics Education 

9 Curriculum and Evaluation Standards for School Mathematics 
9 Science for All Americans 
9 Reshaping School Mathematics: A Philosophy and Framework 

for Curriculum 

Taken together these publications provide a consistent and urgent vision 
that should help the United States restore excellence to mathematics 

Although five examples are presented in this volume, they are certainly 
not the only five possibilities. Appropriate curricula for the twenty- 
first century will necessarily involve a wide variety of strands, reflecting 
both the broad spectrum of mathematical sciences and the individual 
choices of local school districts. We offer these themes not as definitive 
recommendations for curricula but as samples of what is possible, to 
stimulate development of new and imaginative programs that reflect 
the vitality and uses of mathematics. 

Although each essay in this volume is the work of one author, each 
has benefited enormously from advice and critique provided by many 
advisers. Overall, the volume was developed under the auspices of the 
1989 MSEB Curriculum Committee chaired by Henry O. Pollak, re- 
tired assistant vice president of Bell Communications Research. Other 
members of this Advisory Committee included Wade Ellis, Jr. of West 
Valley College; Andrew M. Gleason of Harvard University; Martin D. 
Kruskal of Princeton University; Leslie Paoletti of Choate Rosemary 
Hall; Anthony Ralston of the State University of New York at Buffalo; 
Isadore Singer of the Massachusetts Institute of Technology; and Zal- 
man Usiskin of the University of Chicago. These individuals deserve 
much of the credit for helping shape the volume at its inception and for 
keeping it on track. 

Seven "Sounding Boards" were established by the MSEB to review 
drafts of the essays as the volume progressed one for the overview 
paper, one for each main essay, and one to examine the links with sci- 
ence. For "Pattern" the Sounding Board consisted of Isadore Singer and 
Zal Usiskin; for "Dimension," David Masunaga of the lolani School 


in Honolulu and Jean Taylor of Rutgers University; for "Quantity," 
Harvey Keynes of the University of Minnesota and Alan Tucker of the 
State University of New York at Stony Brook; for "Uncertainty," James 
Landwehr of AT&T Bell Laboratories and James Swift of Nanaimo 
Senior Secondary School in British Columbia; for "Shape," Branko 
Griinbaum of the University of Washington and Paula Fitzmaurice of 
Victor J. Andrew High School in Tinley Park, Illinois; and for "Change," 
Robert Devaney of Boston University and Leslie Paoletti. The Scientific 
Sounding Board which reviewed the entire volume consisted of William 
O. Baker, retired chairman of Bell Laboratories; Maurice Fox, professor 
of biology at MIT; and Gerard Debreau, professor of economics at the 
University of California at Berkeley. 

Many improvements in this volume are due directly to the hard work 
and good ideas of these distinguished Sounding Board reviewers. To be 
fair to them, however, it is important to acknowledge that the authors 
did not always heed the advice proffered by their reviewers; so while 
we are genuinely grateful for their assistance, full responsibility for the 
points of view expressed in this volume rests with the authors. 

Publication of this volume completes the first phase in the work of the 
MSEB to express to the nation a new vision of mathematics education, 
of how a centuries-old curriculum can evolve to meet the challenges of 
the next millennium. Right from the beginning of MSEB in 1985, for- 
mer MSEB Chair Shirley Hill of the University of Missouri at Kansas 
City took up the difficult challenge of forcing mathematicians and math- 
ematics educators to think together about possible new strands for the 
mathematics curriculum. She challenged all of us on the MSEB to seek 
out ideas that may be more appropriate to our computer age than the 
arithmetic-bound structures that we have inherited from previous gen- 
erations for whom calculation was the primary purpose of mathematics. 
This volume is the direct result of Shirley's persistence in emphasizing 
the importance of rooting curricular reform in the emerging practice of 

Coordination and production details have been ably managed by the 
MSEB staff led first by Marcia Sward and now by Kenneth Hoffman. 
Special thanks are due Linda Rosen, who shepherded with unfailing 
good humor all technical aspects of production from the initial plan- 
ning meetings to final details of artwork, copy editing, and production. 
Thanks also are due Jana Godsey whose tenacity and patience were in- 
valuable in collecting the many illustrations for the volume. Much of 
the computer-generated artwork was provided by Thomas Banchoff and 
David Moore, with special support from Davide Cervone, a graduate 
student at Brown University. Finally, throughout the many drafts of the 


different essays, Mary Kay Peterson managed with efficiency all the TgX 
typing and corrections necessary to enable the final text to be produced 
by direct electronic means. 

Lynn Arthur Steen, Editor 
St. Olaf College 



Lynn Arthur Steen 


Thomas F. Banchoff 


James T. Fey 


David S. Moore 

SHAPE/ 139 

Majorie Senechal 

CHANGE/ 183 

Ian Stewart 

INDEX / 223 



"He just saw further than the rest of us." The subject of this remark, 
cyberneticist Norbert Wiener, is one of many exceptional scientists who 
broke the bonds of tradition to create entirely new domains for math- 
ematicians to explore. Seeing and revealing hidden patterns are what 
mathematicians do best. Each major discovery opens new areas rich 
with potential for further exploration. In the last century alone, the 
number of mathematical disciplines has grown at an exponential rate; 
examples include the ideas of Georg Cantor on transfinite sets, Sonja 
Kovalevsky on differential equations, Alan Turing on computability, 
Emmy Noether on abstract algebra, and, most recently, Benoit Mandel- 
brot on fractals. 

To the public these new domains of mathematics are terra incognita. 
Mathematics, in the common lay view, is a static discipline based on 
formulas taught in the school subjects of arithmetic, geometry, algebra, 
and calculus. But outside public view, mathematics continues to grow 
at a rapid rate, spreading into new fields and spawning new applications. 
The guide to this growth is not calculation and formulas but an open- 
ended search for pattern. 

Mathematics has traditionally been described as the science of num- 
ber and shape. The school emphasis on arithmetic and geometry is 
deeply rooted in this centuries-old perspective. But as the territory ex- 
plored by mathematicians has expanded into group theory and statis- 
tics, into optimization and control theory the historic boundaries of 
mathematics have all but disappeared. So have the boundaries of its 


applications: no longer just the language of physics and engineering, 
mathematics is now an essential tool for banking, manufacturing, social 
science, and medicine. When viewed in this broader context, we see 
that mathematics is not just about number and shape but about pattern 
and order of all sorts. Number and shape arithmetic and geometry 
are but two of many media in which mathematicians work. Active 
mathematicians seek patterns wherever they arise. 

Thanks to computer graphics, much of the mathematician's search for 
patterns is now guided by what one can really see with the eye, whereas 
nineteenth-century mathematical giants like Gauss and Poincare had 
to depend more on seeing with their mind's eye. "I see" has always 
had two distinct meanings: to perceive with the eye and to understand 
with the mind. For centuries the mind has dominated the eye in the 
hierarchy of mathematical practice; today the balance is being restored 
as mathematicians find new ways to see patterns, both with the eye and 
with the mind. 

Change in the practice of mathematics forces re-examination of math- 
ematics education. Not just computers, but also new applications and 
new theories have expanded significantly the role of mathematics in sci- 
ence, business, and technology. Students who will live and work using 
computers as a routine tool need to learn a different mathematics than 
their forefathers. Standard school practice, rooted in traditions that are 
several centuries old, simply cannot prepare students adequately for the 
mathematical needs of the twenty-first century. 

Shortcomings in the present record of mathematical education also 
provide strong forces for change. Indeed, since new developments build 
on fundamental principles, it is plausible, as many observers often sug- 
gest, that one should focus first on restoring strength to time-honored 
fundamentals before embarking on reforms based on changes in the 
contemporary practice of mathematics. Public support for strong ba- 
sic curricula reinforces the wisdom of the past that traditional school 
mathematics, if carefully taught and well learned, provides sound prepa- 
ration both for the world of work and for advanced study in mathemat- 
ically based fields. 

The key issue for mathematics education is not whether to teach fun- 
damentals but which fundamentals to teach and how to teach them. 
Changes in the practice of mathematics do alter the balance of priori- 
ties among the many topics that are important for numeracy. Changes 
in society, in technology, in schools among others will have great im- 
pact on what will be possible in school mathematics in the next century. 
All of these changes will affect the fundamentals of school mathematics. 

To develop effective new mathematics curricula, one must attempt 
to foresee the mathematical needs of tomorrow's students. It is the 


present and future practice of mathematics at work, in science, in 
research that should shape education in mathematics. To prepare ef- 
fective mathematics curricula for the future, we must look to patterns 
in the mathematics of today to project, as best we can, just what is and 
what is not truly fundamental. 


School tradition has it that arithmetic, measurement, algebra, and 
a smattering of geometry represent the fundamentals of mathematics. 
But there is much more to the root system of mathematics deep ideas 
that nourish the growing branches of mathematics. One can think of 
specific mathematical structures: 

Numbers Shapes 

Algorithms Functions 

Ratios Data 

or attributes: 

Linear Random 

Periodic Maximum 

Symmetric Approximate 

Continuous Smooth 

or actions: 






or abstractions: 





or attitudes: 



or behaviors: 














or dichotomies: 

Discrete vs. continuous 

Finite vs. infinite 

Algorithmic vs. existential 

Stochastic vs. deterministic 

Exact vs. approximate 

These diverse perspectives illustrate the complexity of structures that 
support mathematics. From each perspective one can identify vari- 
ous strands that have within them the power to develop a significant 
mathematical idea from informal intuitions of early childhood all the 
way through school and college and on into scientific or mathematical 
research. A sound education in the mathematical sciences requires en- 
counter with virtually all of these very different perspectives and ideas. 

Traditional school mathematics picks very few strands (e.g., arith- 
metic, geometry, algebra) and arranges them horizontally to form the 
curriculum: first arithmetic, then simple algebra, then geometry, then 
more algebra, and finally as if it were the epitome of mathematical 
knowledge calculus. This layer-cake approach to mathematics educa- 
tion effectively prevents informal development of intuition along the 
multiple roots of mathematics. Moreover, it reinforces the tendency 
to design each course primarily to meet the prerequisites of the next 
course, making the study of mathematics largely an exercise in delayed 
gratification. To help students see clearly into their own mathematical 
futures, we need to construct curricula with greater vertical continuity, 
to connect the roots of mathematics to the branches of mathematics in 
the educational experience of children. 

School mathematics is often viewed as a pipeline for human resources 
that flows from childhood experiences to scientific careers. The layers 
in the mathematics curriculum correspond to increasingly constricted 
sections of pipe through which all students must pass if they are to 
progress in their mathematical and scientific education. Any imped- 
iment to learning, of which there are many, restricts the flow in the 
entire pipeline. Like cholesterol in the blood, mathematics can clog the 
educational arteries of the nation. 

In contrast, if mathematics curricula featured multiple parallel strands, 
each grounded in appropriate childhood experiences, the flow of human 
resources would more resemble the movement of nutrients in the roots 
of a mighty tree or the rushing flow of water from a vast watershed 
than the increasingly constricted confines of a narrowing artery or pipe- 
line. Different aspects of mathematical experience will attract children 
of different interests and talents, each nurtured by challenging ideas 
that stimulate imagination and promote exploration. The collective 


effect will be to develop among children diverse mathematical insight 

in many different roots of mathematics. 


This volume offers five examples of the developmental power of deep 
mathematical ideas: dimension, quantity, uncertainty, shape, and 
change. Each chapter explores a rich variety of patterns that can be 
introduced to children at various stages of school, especially at the 
youngest ages when unfettered curiosity remains high. Those who de- 
velop curricula will find in these essays many valuable new options for 
school mathematics. Those who help determine education policy will 
see in these essays examples of new standards for excellence. And ev- 
eryone who is a parent will find in these essays numerous examples of 
important and effective mathematics that could excite the imagination 
of their children. 

Each chapter is written by a distinguished scholar who explains in 
everyday language how fundamental ideas with deep roots in the math- 
ematical sciences could blossom in schools of the future. Although 
not constrained by particular details of present curricula, each essay 
is faithful to the development of mathematical ideas from childhood to 
adulthood. In expressing these very different strands of mathematical 
thought, the authors illustrate ideals of how mathematical ideas should 
be developed in children. 

In contrast to much present school mathematics, these strands are 
alive with action: pouring water to compare volumes, playing with pen- 
dulums to explore dynamics, counting candy colors to grasp variation, 
building kaleidoscopes to explore symmetry. Much mathematics can 
be learned informally by such activities long before children reach the 
point of understanding algebraic formulas. Early experiences with such 
patterns as volume, similarity, size, and randomness prepare students 
both for scientific investigations and for more formal and logically pre- 
cise mathematics. Then when a careful demonstration emerges in class 
some years later, a student who has benefited from substantial early in- 
formal mathematical experiences can say with honest pleasure "Now I 
see why that's true." 


The essays in this volume are written by five different authors on five 
distinct topics. Despite differences in topic, style, and approach, these 
essays have in common the lineage of mathematics: each is connected 
in myriad ways to the family of mathematical sciences. Thus it should 


come as no surprise that the essays themselves are replete with inter- 
connections, both in deep structure and even in particular illustrations. 
Some examples: 

MEASUREMENT is an idea treated repeatedly in these essays. Experi- 
ence with geometric quantities (length, area, volume), with arithmetic 
quantities (size, order, labels), with random variation (spinners, coin 
tosses, SAT scores), and with dynamic variables (discrete, continuous, 
chaotic) all pose special challenges to answer a very child-like question: 
"How big is it?" One sees from many examples that this question is 
fundamental: it is at once simple yet subtle, elementary yet difficult. 
Students who grow up recognizing the complexity of measurement may 
be less likely to accept unquestioningly many of the common misuses 
of numbers and statistics. Learning how to measure is the beginning of 

SYMMETRY is another deep idea of mathematics that turns up over 
and over again, both in these essays and in all parts of mathematics. 
Sometimes it is the symmetry of the whole, such as the hypercube (a 
four-dimensional cube), whose symmetries are so numerous that it is 
hard to count them all. (But with proper guidance, young children us- 
ing a simple pea-and-toothpick model can do it.) Other times it is the 
symmetry of the parts, as in the growth of natural objects from repet- 
itive patterns of molecules or cells. In still other cases it is symmetry 
broken, as in the buckling of a cylindrical beam or the growth of a 
fertilized egg to a (slightly) asymmetrical adult animal. Unlike mea- 
surement, symmetry is seldom studied much in school at any level, yet 
it is equally fundamental as a model for explaining features of such di- 
verse phenomena as the basic forces of nature, the structure of crystals, 
and the growth of organisms. Learning to recognize symmetry trains 
the mathematical eye. 

VISUALIZATION recurs in many examples in this volume and is one 
of the most rapidly growing areas of mathematical and scientific re- 
search. The first step in data analysis is the visual display of data to 
search for hidden patterns. Graphs of various types provide visual dis- 
play of relations and functions; they are widely used throughout science 
and industry to portray the behavior of one variable (e.g., sales) that 
is a function of another (e.g., advertising). For centuries artists and 
map makers have used geometric devices such as projection to repre- 
sent three-dimensional scenes on a two-dimensional canvas or sheet of 
paper. Now computer graphics automate these processes and let us 
explore as well the projections of shapes in higher-dimensional space. 
Learning to visualize mathematical patterns enlists the gift of sight as 
an invaluable ally in mathematical education. 


ALGORITHMS are recipes for computation that occur in every corner 
of mathematics. A common iterative procedure for projecting popula- 
tion growth reveals how simple orderly events can lead to a variety of 
behaviors explosion, decay, repetition, chaos. Exploration of combi- 
natorial patterns in geometric forms enables students to project geomet- 
ric structures in higher dimensions where they cannot build real mod- 
els. Even common elementary school algorithms for arithmetic take on 
a new dimension when viewed from the perspective of contemporary 
mathematics: rather than stressing the mastery of specific algorithms 
which are now carried out principally by calculators or computers 
school mathematics can instead emphasize more fundamental attributes 
of algorithms (e.g., speed, efficiency, sensitivity) that are essential for 
intelligent use of mathematics in the computer age. Learning to think 
algorithmically builds contemporary mathematical literacy. 

Many other connective themes recur in this volume, including link- 
ages of mathematics with science, classification as a tool for understand- 
ing, inference from axioms and data, and most importantly the role 
of exploration in the process of learning mathematics. Connections give 
mathematics power and help determine what is fundamental. Pedagog- 
ically, connections permit insight developed in one strand to infuse into 
others. Multiple strands linked by strong interconnections can develop 
mathematical power in students with a wide variety of enthusiasms and 


Newton credited his extraordinary foresight in the development of 
calculus to the accumulated work of his predecessors: "If I have seen 
farther than others, it is because I have stood on the shoulders of giants." 
Those who develop mathematics curricula for the twenty-first century 
will need similar foresight. 

Not since the time of Newton has mathematics changed as much as it 
has in recent years. Motivated in large part by the introduction of com- 
puters, the nature and practice of mathematics have been fundamentally 
transformed by new concepts, tools, applications, and methods. Like 
the telescope of Galileo's era that enabled the Newtonian revolution, to- 
day's computer challenges traditional views and forces re-examination 
of deeply held values. As it did three centuries ago in the transition 
from Euclidean proofs to Newtonian analysis, mathematics once again 
is undergoing a fundamental reorientation of procedural paradigms. 

Examples of fundamental change abound in the research literature 
of mathematics and in practical applications of mathematical methods. 
Many are given in the essays in this volume: 


Uncertainty is not haphazard, since regularity eventually emerges. 

Deterministic phenomena often exhibit random behavior. 

Dimensionality is not just a property of space but also a means 
of ordering knowledge. 

Repetition can be the source of accuracy, symmetry, or chaos. 

Visual representation yields insights that often remain hidden 
from strictly analytic approaches. 

Diverse patterns of change exhibit significant underlying regular- 

By examining many different strands of mathematics, we gain per- 
spective on common features and dominant ideas. Recurring concepts 
(e.g., number, function, algorithm) call attention to what one must know 
in order to understand mathematics; common actions (e.g., represent, 
discover, prove) reveal skills that one must develop in order to do math- 
ematics. Together, concepts and actions are the nouns and verbs of the 
language of mathematics. 

What humans do with the language of mathematics is to describe pat- 
terns. Mathematics is an exploratory science that seeks to understand 
every kind of pattern patterns that occur in nature, patterns invented 
by the human mind, and even patterns created by other patterns. To 
grow mathematically, children must be exposed to a rich variety of pat- 
terns appropriate to their own lives through which they can see variety, 
regularity, and interconnections. 

The essays in this volume provide five extended case studies that ex- 
emplify how this can be done. Other authors could just as easily have 
described five or ten different examples. The books and articles listed 
below are replete with additional examples of rich mathematical ideas. 
What matters in the study of mathematics is not so much which partic- 
ular strands one explores, but the presence in these strands of significant 
examples of sufficient variety and depth to reveal patterns. By encour- 
aging students to explore patterns that have proven their power and 
significance, we offer them broad shoulders from which they will see 
farther than we can. 


1. Albers, Donald J. and Alexanderson, G.L. Mathematical People: Profiles and Inter- 
views. Cambridge, MA: Birkhauser Boston, 1985. 

2. Barnsley, Michael F. Fractals Everywhere. New York, NY: Academic Press, 1988. 

3. Barnsley, Michael F. The Desktop Fractal Design System. New York, NY: Academic 
Press, 1989. 

4. Brook, Richard J., et al. (Eds.). The Fascination of Statistics. New York, NY: Marcel 
Dekker, 1986. 


5. Campbell, Stephen K. Flaws and Fallacies in Statistical Thinking. Englewood Cliffs, 
NJ: Prentice-Hall, 1974. 

6. Davis, Philip J. and Hersh, Reuben. Descartes ' Dream: The World According to 
Mathematics. San Diego, CA: Harcourt Brace Jovanovich, 1986. 

7. Davis, Philip J. and Hersh, Reuben. The Mathematical Experience. Boston, MA: 
Birkhauser, 1980. 

8. Devaney, Robert L. Chaos, Fractals, and Dynamics: Computer Experiments in Math- 
ematics. Reading, MA: Addison- Wesley, 1990. 

9. Dewdney, A.K. The Turing Omnibus: 61 Excursions in Computer Science. Rockvilie, 
MD: Computer Science Press, 1989. 

1 0. Ekeland, Ivar. Mathematics and the Unexpected. Chicago, IL: University of Chicago 
Press, 1988. 

1 1 . Fischer, Gerd. Mathematical Models from the Collections of Universities and Muse- 
urns. Wiesbaden, FRG: Friedrich Vieweg & Sohn, 1986. 

12. Francis, George K. A Topological Picturebook. New York, NY: Springer- Verlag, 1987. 

13. Gleick, James. Chaos. New York, NY: Viking Press, 1988. 

14. Guillen, Michael. Bridges to Infinity: The Human Side of Mathematics. Boston, MA: 
Houghton Mifflin, 1983. 

15. Hoffman, Paul. Archimedes' Revenge: The Joys and Perils of Mathematics. New 
York, NY: W.W. Norton & Company, 1988. 

16. Hofstadter, Douglas R. Godel, Escher, Bach: An Eternal Golden Braid. New York, 
NY: Vintage Press, 1980. 

1 7. Holden, Alan. Shapes, Space, and Symmetry. New York, NY: Columbia University 
Press, 1971. 

18. Huff, Darrell. How to Lie with Statistics. New York, NY: W.W. Norton & Company, 

19. Huntley, H.E. The Divine Proportion: A Study in Mathematical Beauty. Mineola, 
NY: Dover, 1970. 

20. Jaffe, Arthur. "Ordering the Universe: The Role of Mathematics." In Renewing U.S. 
Mathematics: Critical Resource for the Future. Washington, DC: National Academy 
Press, 1984, 117-162. 

21. Kitcher, Philip. The Nature of Mathematical Knowledge. New York, NY: Oxford 
University Press, 1983. 

22. Kline, Morris. Mathematics and the Search for Knowledge. New York, NY: Oxford 
University Press, 1985. 

23. Lang, Serge. MATH! Encounters with High School Students. New York, NY: Springer- 
Verlag, 1985. 

24. Lang, Serge. The Beauty of Doing Mathematics: Three Public Dialogues. New York, 
NY: Springer- Verlag, 1985. 

25. Loeb, Arthur. Space Structures: Their Harmony and Counterpoint. Reading, MA: 
Addison- Wesley, 1976. 

26. Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York, NY: W.H. Free- 
man, 1982. 

27. Moore, David S. Statistics: Concepts and Controversies, Second Edition. New York, 
NY: W.H. Freeman, 1985. 

28. Morrison, Philip and Morrison, Phylis. Powers of Ten. New York, NY: Scientific 
American Books, 1982. 

29. Peitgen, Heinz-Otto and Richter, Peter H. The Beauty of Fractals: Images of Complex 
Dynamical Systems. New York, NY: Springer- Verlag, 1986. 

30. Peitgen, Heinz-Otto and Saupe, Dietmar (Eds.). The Science of Fractal Images. New 
York, NY: Springer- Verlag, 1988. 


31. Peterson, Ivars. The Mathematical Tourist: Snapshots of Modem Mathematics. New 
York, NY: W.H. Freeman, 1988. 

32. Rosen, Joe. Symmetry Discovered: Concepts and Applications in Nature and Science. 
New York, NY: Cambridge University Press, 1975. 

33. Rucker, Rudy. Infinity and the Mind: The Science and Philosophy of the Infinite. 
Boston, MA: Birkhauser, 1982. 

34. Rucker, Rudy. The Fourth Dimension: Toward a Geometry of Higher Reality. Boston, 
MA: Houghton Mifflin, 1984. 

35. Senechal, Marjorie and Fleck, George (Eds.). Patterns of Symmetry. Amherst, MA: 
University of Massachusetts Press, 1977. 

36. Sondheimer, Ernst and Rogerson, Alan, Numbers and Infinity: A Historical Account 
of Mathematical Concepts. New York, NY: Cambridge University Press, 1981. 

37. Steen, Lynn Arthur. Mathematics Today: Twelve Informal Essays. New York, NY: 
Springer-Verlag, 1978. 

38. Steen, Lynn Arthur. "The Science of Patterns." Science, 240 (29 April 1988), 611- 

39. Steinhaus, H. Mathematical Snapshots, Third American Edition Revised and Enlarged. 
New York, NY: Oxford University Press, 1983. 

40. Stevens, Peter S. Patterns in Nature. Boston, MA: Little, Brown & Company, 1974. 

41. Stewart, Ian. The Problems of Mathematics. New York, NY: Oxford University Press, 

42. Stewart, Ian. Does God Play Dice? The Mathematics of Chaos. Oxford: Blackwell, 

43. Tanur, Judith M., et al. (Eds.). Statistics: A Guide to the Unknown, Third Edition. 
Laguna Hills, CA: Wadsworth, 1989. 

44. Tufte, Edward R. The Visual Display of Quantitative Information. Cheshire, CT: 
Graphics Press, 1983. 

45. Wenninger, Magnus J. Polyhedron Models for the Classroom, Second Edition. Reston, 
VA: National Council of Teachers of Mathematics, 1975. 




One hundred and fifty years ago, Friedrich Froebel (Figure 1), the 
inventor of the term "kindergarten," devised a set of "gifts" to intro- 
duce children to notions of geometry in several different dimensions. 
His philosophy was clear: if children could be stimulated to observe 
geometric objects from the earliest stage of their education, these ideas 
would come back to them again and again during the course of their 
schooling, deepening with each new level of sophistication. The rudi- 
mentary appreciation of shapes and forms at the nursery school level 
would become more refined as students developed new skills in arith- 
metic and measurement and later in more formal algebra and geometry. 

In order to capture the imaginations of his young students, Froebel 
presented them with a sequence of wooden objects for their play in the 
Children's Garden. Only later would the lessons of that directed set 
of play experiences be turned into concepts and even later formalized 
into mathematical expressions. The important thing was to introduce 
students to forms that they could apprehend and to encourage them 
to observe and recognize those forms in all of their experiences. In 
this way they could foster the facility of visualization, so important in 
applying mathematics to both scientific and artistic pursuits. 

Froebel began with objects from the most concrete part of mathe- 
matics: balls, cubes, and cylinders. He proceeded to a higher level of 
abstraction by presenting the children with trays covered by patterns of 



FIGURE 1. Friedrich Froebel, inventor of kindergarten, used 
geometric objects to stimulate children's imaginations. 

tiles. Then he moved further into abstraction by introducing collections 
of sticks of varying lengths, to be placed in designs that would ultimately 
be related to number patterns. 

We can recognize some of FroebePs legacy in materials that we find in 
today's kindergarten classrooms. There we still have blocks for stacking 
and tiles for creating patterns on tabletops. Too often, however, these 
"toys" are left behind when children progress into the serious world 
of elementary school. A great many rods are used for arithmetic exer- 
cises, but a student is lucky to see anything two-dimensional between 
kindergarten and junior high school. At that time there might be a brief 
mention of area of plane figures, often merely as an illustration of for- 
mulas for measurement. Then the student must wait until high school 
before any further thought is given to the world of plane geometry. 

Two generations ago the hardy souls who made it through the year of 
formal geometry were permitted to re-enter the third dimension in a still 
more formalized semester of solid geometry. Then curricula changed. 


Three-dimensional topics (along with all of analytic geometry) were sup- 
posed to be incorporated into a single geometry curriculum. All too 
often the solid geometry components were treated merely as supple- 
mentary topics for the interested student who had a bit of leisure time. 
Needless to say, solid geometry quickly evaporated from the standard 
course in geometry. In the present-day rush to prepare students for cal- 
culus before they go off to college, we are systematically shortchanging 
them by ignoring the most practical and useful of all geometry the 
geometry of our own dimension. We now have a special opportunity to 
bring the appreciation of different dimensions back into focus. 

The Dimensional Ladder 

Although our world is three-dimensional, most of our media, as it 
happens, are two-dimensional: blackboards, books, movies, television, 
and computer screens. We all invest a great deal of effort learning how 
to interpret such planar visual information, often in order to help us 
deal with situations in three dimensions. To live in a three-dimensional 
world, we do have to know how two-dimensional shapes interact: their 
behavior provides a necessary prelude to understand fully our own di- 

As it happens, we gain a good deal of insight by investigating the 
geometry of an even lower dimension the line where number and 
geometry intermix in the most intimate and powerful way. The geom- 
etry of the number line translates beautifully into plane geometry, both 
in its classical form and in the analytic geometry of number pairs. The 
momentum that we gain in moving from the first to the second dimen- 
sion can carry us into our home dimension with renewed insight. The 
dimensional analogy is a very powerful tool. 

Here is an exciting theme that is worth recognizing and passing on 
to our students: the momentum that brings us from one to two and up 
into three dimensions does not stop there! The invitation is clear: there 
are other dimensions waiting to be explored. Mathematics is the key to 
the elevator that makes them accessible. 

The fourth dimension, in particular, is one of our nearest neighbors. 
Just as we learn a good deal about our own language and culture by 
studying the language and culture of other countries, so we can be- 
gin to appreciate new things about our own "real" world by seeing 
structures that carry forward to the fourth dimension. Although we 
cannot explore higher dimensions physically, they are accessible to our 
minds and, thanks to modern technology, more and more to our vision 
as well. 


Research into language acquisition indicates that, although any infant 
is capable of learning any language, a child will rather quickly settle into 
the sound patterns of its own particular language, effectively blocking 
development of other possibilities. If a child is not introduced early 
to other languages, he or she will experience much more difficulty in 
learning a second tongue. Might the same be true with respect to math- 
ematical perceptions? If we wait until students have developed a great 
deal of arithmetic sophistication (and a great many misconceptions) be- 
fore we encourage them to think about solid objects and the interaction 
between different dimensions, we may be depriving them of the chance 
to appreciate the full power and scope of geometry. 

Giving Geometrical Gifts 

Objects should always be nearby. Awareness of space and volume 
should be a continuing part of mathematical experience in school at 
all levels. Refinements such as measuring quantities and relating them 
with formulas will come in good time. But they should come well after 
the time when a child first becomes aware of different dimensions of 
measurement. Too often, the first time a student is encouraged to think 
about what volume means is the same day that he or she is given a 
formula for the volume of a sphere or a cone. To encourage fluency 
in the language of geometry, we need a good deal more "pre-geometry" 
throughout the school experience, and that should include "pre-solid" 
as well as "pre-plane" geometry. 

Froebel and his colleagues created geometrical gifts from materials 
available to them, primarily wood, paper, and clay. Today we have the 
means to improve on the gifts in many ways with plastic and Velcro, 
with tape and magnets, not to mention with the powerful computer 
graphics. The educator's term "manipulatives" classroom materials 
takes on new meaning when we can put in front of a young student a tool 
to manipulate not only simple forms but also the very geometry of higher 
dimensional space. If we care about educating our children toward the 
perception of space, we should create truly stimulating manipulatives 
geometrical gifts for our day. 


Many students never learn about volumes because they do not make it 
past plane geometry. Those who do often reach calculus by a head-long 
rush that leaves little or no time for the kind of geometrical thinking on 
which calculus thrives. Calculus is not the time when students should 
be doing their first serious thinking about geometry. Rather it should be 



FIGURE 2. Water in a cylinder ex- 
actly fills three cones whose base and 
height are identical to the base and 
height of the cylinder. 

the culmination of years of consideration of increasingly sophisticated 
geometrical topics. When a student finally sees the full justification of 
the formula for the volume of a cone or a sphere, it should be a peak 
experience, fulfilling a promise implicit in all the experiences he or she 
has had with cones and spheres all the way through school, beginning 
in kindergarten. 

FroebePs young students spent a great deal of time pouring water 
and sifting sand. Differently shaped containers held different amounts, 
so a student would gradually learn common relationships without even 
thinking of writing them down. For example, how many conical cups 
can be filled from the water in a cylindrical cup with the same height 
and the same base? With a rack of such cups (Figure 2), any student 
can perform the experiment. The cylinder fills three cups. 

We can test this over and over again with different heights and dif- 
ferent circular tops. Only later, after the student is familiar with the 
language of fractions, need this relationship be stated in terms of one 
volume being one-third of another. Still later, that relationship can be 
expressed by a formula: the volume of the cone is one-third the area of 
the base multiplied by the height. 

By this time that relationship should already have been observed in 
other shapes. Three square-based pyramids can be filled with the sand 
from one square prism of the same base and height (Figure 3). Even 
if the base is irregular, this relationship is true. We don't even have to 
have the center of the cone over the center of the base, assuming that 
the base even has a center! All this understanding can take place before 
the student has even seen a fraction, let alone a number like x. 

FIGURE 3. Water in a prism exactly fills three pyramids whose 
base and height match those of the prism. This relation holds even 
for prisms and cones with irregular bases and can be discovered 
by young children just by pouring water or sand. 



FIGURE 4. Pouring water can also 
verify Archimedes' theorem: the vol- 
ume of three spheres equals the vol- 
ume of two cylinders whose radius and 
height match those of the spheres. 

A bit more subtle and even more impressive is the relationship that 
was symbolized on the gravestone of Archimedes: if a ball fits precisely 
inside a circular cylinder, then the volume of the ball is two-thirds the 
volume of the cylinder. To illustrate this we can show that three spheres 
can be filled with the water from two cylinders that encase the spheres 
(Figure 4). Volumes of irregularly shaped objects can be found by seeing 
how much water they displace when they are completely submerged. 
This leads naturally to the notion of density, as a weight-to-volume 

The notion of area can be introduced by working with volumes. By 
using a collection of shallow pans, all of the same height, children can 
compare their volumes and relate them to the areas of their bases. The 
height dimension is "washed out" if it is the same in all cases. In this 
way it is easy for children to see that the area of a right triangle is half 
the area of the associated rectangle and that the area of a scalene triangle 
is half the area of three different associated parallelograms (Figure 5). 

FIGURE 5. By pouring water into shallow pans, children can 
readily compare the areas of different geometric figures. 

FIGURE 6. Four right triangles in a 
square frame reveal a proof of the 
Pythagorean theorem: the square on 
the hypotenuse equals the sum of the 
squares on the legs of the right triangle. 



We can work as well with tiles of uniform thickness, as Froebel did in 
his kindergarten gifts in the last century. The relation between the area 
of a parallelogram and the area of a rectangle can be appreciated at a 
very early stage by students who actually manipulate physical objects. It 
isn't necessary to wait until students have learned about square roots be- 
fore they can see an illustration of the Pythagorean theorem (Figure 6). 
Children who play with geometric puzzles that illustrate decompositions 
will find it much easier later on to appreciate formal results. 

Decomposition Models 

One of the most beautiful results that can be illustrated by blocks 
is the fact that a cube can be decomposed into three identical pieces 
meeting along a diagonal of the cube (Figure 8), just as a square is 
decomposed into two congruent triangles by a diagonal line (Figure 7). 

FIGURE 7. The diagonal subdivision 
of a square into two congruent triangles 
serves as a prelude to a similar decom- 
position in three dimensions. 

FIGURE 8. The diagonal decomposi- 
tion of a cube into congruent pyramids 
can be illustrated by blocks built from 
corresponding templates. 


FIGURE 9. The diagonal decomposition of a rectangular solid 
yields three pyramids of different shapes but the same volume. 

Decomposition models illustrate deeper ideas than do comparisons of 
volumes since they not only demonstrate relationships but also show 
why these relationships hold. Students should eventually come to see 
that all geometric relationships are based on reasons. 

This particular decomposition property of the cube can be a bit mis- 
leading because it doesn't quite work for other rectangular solids. Al- 
though a diagonal always decomposes a rectangle into two congruent 
triangles, the diagonal decomposition of a rectangular solid will usually 
not produce three congruent pyramids (Figure 9). The three pyramidal 
parts will all have the same volume but not the same shape. This can 
be seen by pouring sand into plastic pyramid containers, but greater 
insight comes from a different model playing cards. 

Think of a pyramid constructed of thick rectangular cards stacked 
above the base. If we double the thickness of each card in the stack, 
then the base stays the same while both the height and the weight of 
the stack (and therefore its volume) also double. If we keep the width 
and the thickness of each card the same and double the length, then the 
volume also doubles. Doubling any single dimension causes the volume 
to double: in general, multiplying a single dimension by any number 
will multiply the entire volume by that same number. 

This procedure enables us to obtain the volume of any pyramid formed 
by a diagonal decomposition of a rectangular solid that is, of any pyra- 
mid with a rectangular base whose top vertex is directly over a corner of 
the base. Further work with pyramid-shaped blocks will quickly show 
that any pyramid with a rectangular base can be built up from pyra- 
mids of this special type, all with the same height. Taken together, 
these demonstrations show why, in general, the volume of a pyramid 
with a rectangular base is one-third the volume of the right rectangular 
prism with the same base and height. 

Experiments with stacks of cards or thin rods can lead easily to a pow- 
erful idea known to mathematicians as Cavalieri's principle for shear 
transformations. First observe how the same set of rods that fills a paral- 
lelogram will also fill a rectangle with the same base and height. Hence 

FIGURE 1 0. The same set of rods that forms a rectangle can also 

form a parallelogram of the same dimensions. Hence the areas of 
the rectangle and the parallelogram must be the same. 

their areas must be equal (Figure 10). The same principle works in 
space as well as in the plane. The same pack of cards that fills a straight 
box can fill a slanted box with the same base and height. Similarly, 
an off-center pyramid can be approximated with the same collection of 
square cards that approximate a centered one (Figure 1 1). 

Students who explore models of pyramids with sets of blocks and 
stacks of cards throughout their early school years are certainly more 
likely to understand and appreciate the formal proofs presented for such 
theorems in calculus classes; students who have never thought about 
properties of volumes until they arise in calculus will not get nearly as 
much out of their experience. We now spend a great deal of effort getting 

FIGURE 11. The same set of cards that 
forms an off-centered pyramid can be rear- 
ranged to form a centered pyramid of the same 
base, height, and volume. 


students ready for the algebraic techniques needed for advanced mathe- 
matics. We should be just as concerned for their geometric preparation 
as well. 

Pyramid Problems 

Many children are fascinated by the great pyramids of Cheops. These 
only surviving wonders of the ancient world were mathematical chal- 
lenges to their creators, and they remain challenging today. School study 
of the monuments of ancient Egypt can be a source of mathematics 
problems of all sorts, from the most elementary considerations of shad- 
ows to the most sophisticated achievement of early mensuration the 
volume formula for the frustrum of a truncated pyramid. 

Children can decide how to make models of the pyramids. A pile of 
dry sand or wet sand on a square base provides one example. Models 
in clay provide another. Students can experiment with different sizes of 
triangles to see what shapes of pyramids result. 

Other monuments of different shapes provide similar exercises in 
measurement and challenges for construction. What about the burial 
mounds of American Indians or other cone-shaped structures? What 
about Mayan pyramids, with their step-like structure? What about 
Babylonian ziggerats, or pagodas? Each structure provides distinctive 
features that lead to interesting mathematical questions, which the stu- 
dents themselves can formulate and explore. 

A key mathematical notion that arises naturally in the study of mon- 
uments is similarity, expressed both algebraically in ratio or proportion 
and geometrically in shadows and scale diagrams. Consider the follow- 
ing story: 

My friend Ambrose sent a snapshot of his trip to Egypt. He 
is standing next to an obelisk and I can see that his shadow is 
about one-fourth as long as the shadow of the obelisk. That's a 
pretty big column, over 24 feet high. I know that because my 
friend is 6 feet tall. There is a pyramid in the picture too. I 
can see that its shadow is falling just past the edge of the base. 
What additional information would I need in order to figure out 
how high the pyramid is? How can I measure the angle that the 
slanting side of the pyramid makes with the ground? 

Such questions can be discussed at an informal level long before the 
students deal with triangles formally in geometry and trigonometry. 

Thinking about the pyramids can show how problems in different di- 
mensions can illuminate each other. Using the principle of similarity, 
students can easily calculate the volume of an incomplete pyramid (Fig- 
ure 12), one of the most important problems in Egyptian mathematics. 



FIGURE 12. An incomplete (or truncated) 
pyramid poses a challenge to find its volume. 


FIGURE 1 3. By thinking of a trapezoid as an incomplete trian- 
gle, we can find a way to calculate its area that can also be used in 
three dimensions to find the volume of an incomplete pyramid. 

Begin with the analogous problem in the plane: the trapezoid viewed 
as an incomplete triangle (Figure 13). We know the quantities a, b, and 
h, and we want to find the area. Assuming that the trapezoid is not a 
parallelogram, we can complete the figure to a triangle with height that 
we call x. By observing that the large and small triangles are similar, we 
see that x/a = (jc + h)/b. Hence bx = ax + ah, so x = ha/(b - a) and 
x + h = hbj(b a). We then get the familiar formula for area of the 
trapezoid in a new way, as the difference of the areas of two triangles: 

= (l/2)hb 2 /(b -a)- (l/2)ha 2 /(b 
=:(l/2)h(b 2 -a 2 )/(b-a) 

The same method enables one to calculate the volume of the incom- 
plete pyramid (Figure 14). We are given the height h of part of the 
pyramid and the side lengths a and b of the top and bottom squares. If 
the height of the large pyramid is (x -f h), then its total volume will be 


FIGURE 14. By completing the incomplete 
pyramid, its volume can be calculated as the 
difference of the volumes of two similar pyra- 

By similar triangles, x/a = (x + h)/b. So, as in the planar case, 
x = ha/(b - a) and x + h = hb/(b - a). Therefore the volume of 
the incomplete pyramid is 

ab + a 2 ). 

This formula, which was detailed in a papyrus from 1800 B.C., rep- 
resents a high point in the geometry of the ancient world. It can be 
appreciated by any student who reaches the level of first-year algebra. 
Truly enterprising students can conjecture the formula for the volume 
of an incomplete pyramid in the fourth dimension or in higher dimen- 

Cylinders and Discs 

The volume of water in a circular cylinder is a little more than three- 
quarters of the volume of the rectangular box in which the cylinder 
just fits (Figure 15). If we pour the water from the cylinder into box- 
shaped containers of the same height, with square base whose side equals 
the radius of the cylinder, then we can fill three such boxes and still 
have some water left over. Experiments with different cylinders and 
related boxes will quickly show that this pattern works for cylinders 
of any radius or height. The same ratio, of course, relates the area 
of a circle to its circumscribing square. Because children can measure 
poured quantities more easily than painted areas, it may be easier for 
them to grasp this fundamental ratio first in terms of volume and then 
subsequently in terms of area. 



FIGURE 15. A set of cups containing a circular cylin- 
der matched to four rectangular boxes of the same height 
whose bases form a square that encloses the circular base 
can be used to show that the volume of the cylinder 
is just a little bit more than the volume of three of 
the boxes. Hence the area of the circular base is 
just a bit more than three-quarters of the area of the 
corresponding square. 

The idea of perimeter can be introduced by using a string or a belt, 
unmarked at first. The distance around a square tile is four times the 
length of the side of the tile, regardless of the size of the tile. If one 
circular disc has a radius twice that of another, then a string around 
the larger will fit twice around the smaller. A string around a disc will 
go around a square with sides equal to the radius a little more than 
three times. The crucial fact that the ratio of the circumference of 
the disc to the perimeter of the square is the same as the ratio of the 
volume of the cylinder to the volume of the surrounding box would be 
established only much later. But the fundamental idea that there is a 
fixed ratio between the perimeter of the disc and the perimeter of a 
square is something that every child should appreciate, long before any 
mention of the mysterious number n. 

The relation between the area and circumference of a circle can be 
easily seen by cutting a circle like a pie and reassembling the pieces into 
a nearly rectangular shape. The area of a disc turns out to be equal 
to the area of a rectangle-like region with one side equal to the radius 
and the other equal to half the circumference (Figure 16). Subdividing 
the disc into more slices would make the correspondence even more 
exact. (Much later students will appreciate the limit concept hidden 
in this demonstration.) Unfortunately, there seems to be no such nice 
correspondence between the volume of a sphere and the volume of a 
rectangular box. 



FIGURE 16. By slicing a circle into thin pie-shaped pieces and 
reassembling them into a rectangular-shaped region, children can 
readily see that the area of a circle is the radius (the height of the 
reassembled rectangle) times half of the circumference (the width 
of the rectangle). 


Children in FroebePs kindergarten played with cubes and with subdi- 
vided cubes, squares and subdivided squares, and rods and subdivided 
rods (Figure 17). Eight small cubes fit together to form a large cube, 
twice as long, twice as wide, and twice as high. Four square tiles fit 
together to form a large square, twice as long and twice as wide. Two 
thin rods form a rod twice as long as the original. 

Children at all levels can explore similar exercises. Here is a small 
cardboard box filled with sand, wrapped in paper, and tied with string. 
Here is another box twice as long, twice as wide, and twice as high. 
How much more string do we need to tie it, or paper to cover it, or sand 
to fill it? It isn't necessary to have the ability to measure length or area 
or volume in order to experiment and find the answers: twice as much 
string, four times as many sheets of paper, eight times as much sand. 

FIGURE 17. Nested cubes, squares, 
and rods illustrate the fundamental 
property of doubling factors: they rep- 
resent the power of 2, depending on 


These perceptions about changes of scale can take place even before the 
child has much experience with multiplication, and they can reinforce 
understanding of arithmetic processes. 

Growth Factors 

Children who first encounter changes of scale in the lower grades will 
recognize much later, when they learn about exponential notation, that 
doubling the size in dimension three leads to an increase in the volume 
of a factor of 2 3 , whereas doubling the size of a two-dimensional square 
increases its area by 2 2 . Whatever it might mean to have a box in 
four dimensions, exponents make very clear a pattern of doubling that 
predicts its size will increase by 2 4 . 

Each dimension, therefore, corresponds to its own growth exponent. 
A surprising fact is that there are geometric objects whose growth expo- 
nents are not whole numbers. These strange objects, which have a kind 
of "fractional dimension," are examples of a fascinating collection of 
geometric patterns known as "fractals." Since the creation of fractals 
usually requires a process that is applied an infinite number of times, it 
is only with the advent of modern computer graphics that it has been 
possible to carry out the experiments necessary to explore them effec- 

One of the earliest examples of a fractal was invented long before 
computers by the Polish mathematician Waclaw Sierpinski. The first 
step in creating Sierpinski's figure is to remove a small triangle from the 
middle of a large one. The second step is the same as the first: remove 
the middle of each of the remaining triangles. Repeat this over and over 
again to obtain what is known as the "Sierpinski gasket" (Figure 18). 

What's remarkable about Sierpinski's gasket is that doubling its size 
produces a figure that is composed of three copies of the original fig- 
ure. This is very strange, because our experiments with tiles and cubes 
show that doubling factors are always powers of 2: if we double the 
size of something of dimension one, we get two copies of the origi- 
nal, whereas if we double the size of something of dimension two, we 
get four copies of the original. The Sierpinski gasket, therefore, must 
have a dimension somewhere between one and two hence a fractional 
dimension. (Specifically, its dimension is the number d with the prop- 
erty that 2 d = 3; this number d is the logarithm of three to the base 
two, namely 1.5849....) 

Fractals can be used to motivate a large number of mathematical 
discussions. Since they arise as a result of an infinite process, they can 
be discussed in relation to geometric series or repeating decimals. The 
unusual doubling properties of fractals give a geometric interpretation 




FIGURE 1 8. This infinitely punctured triangle, known as Sierpin- 
ski's gasket, comprises three half-size copies of itself not two or 
four as one would expect if its dimension were one or two. Hence 
it has a fractional dimension in between one and two. 

for the logarithm to base two. Other fractal processes lead to figures 
like the Mandelbrot set, including some of the most striking examples 
of mathematical art. 5 

Rates and Averages 

One of the most important skills we can give our students is the abil- 
ity to interpret data geometrically. The geometry of area and volume 
can help students understand concepts like rates, accumulations, and 
average value. Here are three simple examples that illustrate this point: 

A driver travels at 40 miles per hour for 1 hour, then at 46 miles 
per hour for 2 hours. How far does she travel, and what was her 
average speed? 




FIGURE 19. A bar graph geometrizes 
data from three similar problems and 
shows visually how the average corre- 
sponds to the height of a single rectangle 
with the same base and the same total 

* A designer makes $40,000 a year for 1 year and then $46,000 for 
the next 2 years. What were his total earnings for that period, 
and what was his average salary? 

A fish tank is filled to a depth of 40 centimeters and two identical 
tanks are filled to a depth of 46 centimeters. What is the average 
depth of the water in the tanks? 

All of these problems involve the same calculation, and all can be 
illustrated on the same diagram (Figure 19). In each case the total 
accumulation can be interpreted geometrically as the area of three rect- 
angles. The average will be the height of a single rectangle with the same 
base and the same total area. It is also possible to graph the accumula- 
tion in a way that indicates exactly how many miles had been covered 
(or how much money had been earned) by a given time (Figure 20). 




'(3, 132) 

r (2,86) 

r (l,40) 



FIGURE 20. A linear graph displays the accumulation from three 
problems, indicating total miles covered or dollars earned. The 
relation between the corresponding bar and linear graphs is a 
precursor to calculus. 



Each bar graph representation of rates (which mathematicians call 
a step function) leads to an accumulation graph formed from straight 
lines (i.e., a polygonal function). The process of finding the rate from 
the accumulation leads ultimately to the differential calculus, and find- 
ing accumulations from rates leads to the integral calculus. Although 
it is certainly not necessary for students to realize this connection as 
they develop their understanding of speeds and distances or salaries 
and earnings, every student can benefit from this type of mathematical 
experience both as preparation for calculus and as preparation for life. 

Drawing Cubes 

All children in Froebel's kindergarten practiced drawing. They played 
with drawing on one level while they learned on another. They learned 
to observe spheres, cylinders, and cubes; ultimately they learned to draw 
what they saw. In our day there is not as much emphasis on drawing, so 
we miss opportunities to develop the ability of our students to visualize 
geometrical relationships. 

The most common way of representing a cube in most books is to draw 
a square, then translate it along an oblique axis (usually at a 45 inclina- 
tion) and then to connect corresponding points (Figure 21). Although 
this is a perfectly valid representation of the structure of a transparent 
cube, no view of a cube actually looks like this image. Whenever we 
look at a cube, if one face appears as a square, then we must be looking 
directly toward that face; in this case the opposite face will be directly 
behind the face we see and not off to the side as it is in the traditional 
drawing. This is true whether we use a straight-down "orthographic" 
projection or foreshortening (Figure 22), where the back face appears 
smaller than the front. 

Another popular method of drawing uses "isometric projection," 
which expresses three edges of a cube as segments of equal length 

FIGURE 21. The typical representation of a cube 
as two identical squares with edges connected is 
quite unreal since no cube can ever appear just this 



FIGURE 22. Two correct views of a cube are given by the 
"orthographic" projection (looking straight down) on the left or 
a foreshortened projection (on the right). 

FIGURE 23. The symmetric "isometric" view of a cube, both 
solid and transparent, arise by looking at one corner at a 45 angle. 
Then the opposite corner lies directly behind the front corner, so 
only seven vertices are distinguished in this view. 

FIGURE 24. Two views of a cube in general position: ortho- 
graphic (on the left) and one-point perspective (on the right). 

meeting at 120 angles (Figure 23). This method has the disadvantage 
that two vertices of the cube are represented by the same point. 

If we wish a more general image of a cube, we must draw each face 
as a non-square parallelogram (using a straight-on projection) or as a 
trapezoid (if we use one-point perspective) (Figure 24). The straight-on 
(or orthographic) projection is particularly easy to draw since the picture 
of a cube is completely determined once the position of the edges at one 



FIGURE 25. In an orthographic drawing, parallel lines in the 
cube are rendered as parallel lines on the page. Here the full 
orthographic view of a cube is determined by the orientation of 
the three edges at any corner. 

corner is specified. In an orthographic projection, parallel edges of the 
cube appear as parallel edges in the image, so we can easily complete 
the picture once we know the position of the three edges at any corner 
(Figure 25). 

Once we know how to represent a three-dimensional object on a two- 
dimensional page or computer graphics screen, we can go on to a much 
more complicated exercise, that of drawing a four-dimensional analogue 
of a cube, called a hypercube or tesseract. Many students encounter the 
idea of a four-dimensional cube in science fiction or fantasy literature, 
such as Robert Heinlein's story . . . and He Built a Crooked House 10 or 
Madeleine L'Engle's A Wrinkle in Time 14 or Edwin Abbott Abbott's 
Flatland. 1 

FIGURE 26. By adding a fourth direction to the traditional 
three-line corner that represents three-dimensional space, we lay 
a foundation for drawing four-dimensional objects. It shows the 
direction in which to move a cube to form a four-dimensional 



Usually a hypercube is constructed by moving an ordinary cube in 
a direction perpendicular to our space. Although we cannot actually 
achieve such a motion, we can still draw a picture of what such a con- 
struction would look like when the image is projected to a plane (Figure 
26). We first draw the cube determined by three of the edges, then move 
a copy of the cube along the fourth direction and connect corresponding 

The same procedure enables us to design a three-dimensional model 
of a four-dimensional cube, using sticks attached by clay balls (as sug- 
gested in the last century by Froebel) or more modern materials like 
drinking straws threaded together with yarn, or some standard building 
sets. Once again, the full image of a straight-on projection is determined 
as soon as we specify the four edges coming out of a point (Figure 27). 

Just as a foreshortened view of a cube looks like a square within a 
square with corresponding corners connected, so the analogous fore- 
shortened view of a hypercube looks like a "cube within a cube" with 
corresponding corners connected (Figure 28). 

FIGURE 27. The completed hypercube formed by connecting corresponding vertices on 
two copies of a cube. 

FIGURE 28. A foreshortened view of a hypercube, imagined as a cube within a cube 
with corresponding corners connected. 



FIGURE 29. By rotating a sphere on which an equator has been 
drawn, it is easy to see that the images of a circle are always some 
type of ellipse. 

FIGURE 30. To draw cylinders and 
cones, one begins with an ellipse that 
represents a suitable perspective as the 
circular base. 

A cube looks different from different perspectives. A sphere on the 
other hand always looks like a disc. Any way we look at it, it looks 
the same. If we mark an equator, then various views give images that 
are ellipses in different positions (Figure 29). Students also need to be 
aware of the basic principles of drawing these fundamental forms. It is a 
fact that a circle always looks like an ellipse, including the extreme case 
where the ellipse is still a circle or where it degenerates into a doubly 
covered straight-line segment. Observing this fact makes it easier to 
draw convincing cylinders and cones (Figure 30). 

Modern computers are fast enough to produce a sequence of images 
showing different views of a rotating cube or hypercube, giving the il- 
lusion of a three-dimensional object. This process is very familiar to 
today's students who have grown up with computer-animated special 
effects and television commercials. We can make use of this experience 
to give students new appreciation for mathematical forms. As interac- 
tive programs become more widely available, students of all ages can 
have unprecedented opportunities, never before possible, to manipulate 
and explore geometric forms in three and higher dimensions. 


One of the most important insights we can transmit to students at all 
levels is the utility of coordinate descriptions both to specify locations 
and to give instructions. Examples of coordinates can be made available 
at every stage of a child's development. There is no best way to develop 


understanding of different coordinate dimensions. You don't have to 
learn the first dimension completely before going into the second and 
then the third (and beyond). The invitation to examine coordinates 
from a dimensional standpoint is available at all times: we only have 
to make students aware of what they are seeing. Although experiences 
of different dimensions are always present, it is useful for our present 
analysis to separate phenomena according to the number of coordinates 
needed to locate a position or give an instruction. 

Number Lines and Circles 

Even at a very early age children can understand the significance of 
addresses. Anyone can appreciate the ordinary algorithm used for find- 
ing a specific location in terms of its street address: first go to the street, 
then find the number of some building. If it happens to be the one 
you are looking for, you are done. If not, go to a nearby building and 
check its number. If it is closer to the one you want, keep going in that 
direction. If it is farther away, go in the other direction. Stop when you 
get to the number you want. 

Discussion of even this simple algorithm illustrates a number of im- 
portant topics. We identify a location by a specific number, and we 
move along a one-dimensional path, in one direction or another, to get 
from one position to another. After the basic procedure is understood 
one can add refinements such as whether the address is on the even or 
odd side of the street. Estimating the distance one has to travel in order 
to get from one location to another is another refinement, leading to 
the geometric interpretation of subtraction as well as to the notion of 
absolute value. 

Early exercises can take place on a number line with positive addresses 
or on real streets in a scavenger number hunt. Later the same notions 
can be used for scales with negative values, like temperature, where the 
vertical orientation of the thermometer emphasizes the directionality. 
"What happens when the temperature goes from 65 degrees to 40 de- 
grees?" "It goes down by 25 degrees." Such observations can take place 
far in advance of introducing signed numbers. 

Many cities use directional addresses in their street plan (e.g., in New 
York City there is both a West and an East 42nd Street). In this case 
the algorithm to find a building from knowledge of its address is slightly 
different but still easy enough to discuss at an elementary school level. 
The distance between two addresses on the same side is determined 
as usual, while the distance between two locations on different sides is 
the sum of their addresses. No memorization is required for such a 
statement! Signed numbers do not have to be mysterious. 


A similar one-dimensional algorithm works in setting a clock, wheth- 
er it is analogue or digital, depending on whether or not one can go 
backward as well as forward. Setting a watch is slightly different from 
finding a street address, even on a curving road that does not come 
back on itself. On a circular drive, however, the problem of locating 
a specific address is analogous to the problem of setting a watch: you 
can go in either direction and ultimately arrive at your destination. Of 
course one direction might be much easier than the other. 

The problem of deciding on a strategy for locating an address on a 
circular drive is a good example of the kind of multistep problems that 
students should learn to attack. In this example, as in many others, 
there is no single answer there are several strategies that will achieve 
the same result. The person facing the situation must decide first what 
the choices are and then what might be the advantages of each. The aim 
of minimizing effort is very easy to understand, easier than minimizing 
cost measured in money or some other quantity. 

One-dimensional examples require just one number to locate any 
point. Directions for moving from one position to another are also 
one-dimensional: "Go three houses to the right" or "Go around coun- 
terclockwise five spaces" or "Go halfway around the circle to the oppo- 
site point." This last sort of instruction depends on the size of the circle 
and can form the beginning of an appreciation of angular measure. 

Setting a clock, whether analogue or digital, provides an excellent ex- 
ample of "wrap-around." This phenomenon can also be viewed on a 
linear scale, for example, on the selector of many car radios. In many 
analogue devices the moving indicator stops at the extreme left or ex- 
treme right, while in the digital versions the indicator simply goes from 
the top value to the bottom. Finding a particular radio station then 
presents two different sorts of problems depending on the nature of the 
radio selector. 

The dimensionality of gauges is an important concept that arises over 
and over again in mathematics as well as science. As students become 
more sophisticated in the kinds of numbers they use, they can introduce 
fractions or decimals into number lines and number circles. Locating a 
telephone pole along a road in a rural area requires a different kind of 
address, using fractions or real numbers representing actual distances. 
The numbers become more complicated, but the procedures remain the 

Locating objects or addresses in a one-dimensional world can be ac- 
complished efficiently by the bisection algorithm (or the variation of it 
that divides each interval decimally), a procedure with almost universal 
significance that is related, for example, to the informal technique used 
to find phone numbers. First you make a guess to divide your problem 


into two parts by opening the phone book or by picking a number. 
Then you compare your guess with what you want and make a new 
guess that is in whichever part (above or below your first guess) that 
contains what it is that you are looking for. 

A similar scheme can be used to find the "address" of the length of 
the diagonal of a square without requiring a calculator with a square 
root key. Finding the decimal equivalent of a fraction can be viewed as 
a more sophisticated version of the one-dimensional address problem. 
If we want to find 3/17, we can multiply different decimals by 17 to see 
if the product is bigger or smaller than 3. All decimals get put into one 
category or the other: it never comes out even. For 3/16 on the other 
hand one decimal does come out even, so there is a fixed location on 
the decimal line for the solution to this problem. 

Lengths and Perimeters 

The fundamental geometry problem for one-dimensional phenomena 
is the determination of distance along a path. Key examples include 
calculation or comparison of perimeters of curves and polygons. There 
is one geometric number n that all students should learn to under- 

Despite its universal significance, most people do not know how to 
answer when you ask what n is. Most lay persons respond with a numer- 
ical estimate, 3.1416 or 22/7, without knowing in either case whether 
this approximation is too large or too small. Mathematicians will give a 
definition in terms of a geometric property, usually something like "the 
ratio of the circumference of a circle to its diameter" or "the ratio of 
the area of a disc to the area of a square with side equal to the radius." 
The fact that these two ratios are the same is, of course, a major theo- 
rem of mathematics. One can get a tremendous amount of mileage out 
of a continuing discussion of the estimation of n, from the first time a 
kindergarten student realizes that the belt around a can reaches a little 
more than three times across the top, to second-semester calculus where 
one studies integrals for arc length. 

Finding the circumference of a circle is a one-dimensional problem, 
so its answer should have a representative on the number line. But 
where is it? How can we determine whether or not a given number is 
less than this length or greater? Comparisons with the circumference of 
circumscribed and inscribed polygons is an effective strategy for dealing 
with these questions. Although such comparisons cannot determine n 
exactly, they can convincingly show whether 22/7 is slightly above or 
slightly below n. 


Certain counting games are especially important for developing in 
children facility in the arithmetic of algebraic quantities. Students can 
choose instruction cards saying "move forward two spaces" or "back 
three" (F2 or B3), and they can follow the instructions with counters. 
Then they can be asked to trade two cards for a single card that accom- 
plishes the same effect. By considering double or triple jumps, they gain 
experience with the idea of multiplying a signed number by a positive 
integer. The variations in the game are manifold. The operation of 
taking up three B4 cards from one's hand is the same as taking up one 
B12; putting down three B4 cards is the same as taking up one F12. 
One might introduce a symbolism: P3B4 = "Put down three B4 cards," 
which is the same as T3F4 = "Take up three F4 cards." Similarly, PBS 
= TF5 and PF2 = TB2, yielding a complete algebra of transactions. 

The pedagogical trouble with signed numbers is that we use them 
both for locations and for operations. The rule that "the product of 
two negative numbers is positive" is one of the earliest stumbling blocks 
that convince many students that mathematics means memorizing, not 
reasoning. Appropriate experience with counting games can restore in- 
tuition to the rules of negative numbers. Board games help students 
appreciate the value of scoring, first with simple addition (especially 
where movement depends on the throw of a pair of dice) and later in 
more complicated games where the score can be positive or negative. 
Scoring experiences are generally one-dimensional. 

Planes and Surfaces 

Children should become skilled in both following and giving direc- 
tions. Any child should learn how to direct a person from one part of 
the school to another and perhaps to describe the neighborhood of the 
school. Although the algorithm for getting from one street address to 
another in an actual town might be quite complicated, an ideal town 
has a simpler structure. We can imagine a sequence of imaginary towns 
with different dimensional properties a frontier town all stretched out 
along a single street or a village laid out on a rectangular plot. A model 
village could stimulate a good deal of the discussion, while a grid on 
which children could design their own town would allow for more vari- 

No matter what the streets are named, we can still give directions on a 
grid by saying: "Go right two blocks, then turn left and go three blocks." 
For persons with a clear orientation, the instructions can be varied: "Go 
east two blocks, then north three blocks." The first instruction depends 
on the direction that the person is facing, and the second does not. 


If the map of a village is hanging on a wall, we can use the natural 
coordinate directions: "Go right two blocks and up three." Certain pairs 
of instructions can then be combined: "Go left two, then up three" and 
"Go left three and down five" combine to give "Go left five and down 
two." By playing this game with cards, we can easily introduce the 
operation of adding ordered pairs and even of multiplying numbered 
pairs by positive integers. If we introduce "put" and "take" operations, 
we can extend the one-dimensional algebra of signed numbers to an 
algebra of two-dimensional quantities. 

Notice that this algebra of instructions does not require the use of 
coordinates in the plane. The exercise carries additional value when ad- 
dresses are given in terms of street numbers or compass directions. For 
one thing, this avoids the complications caused by negative numbers. 
To go from E3N4 to E7N2 requires a move of E4S2. The correspon- 
dence between this commonsense approach and the algebraic statement 
(7, 2) - (3, 4) = (4, -2) is something that can come much later in a stu- 
dent's mathematical development. There are a great many people who 
are confused by negative numbers. They shouldn't be. 

"Taxicab geometry" provides an effective variation on the use of di- 
rectional instructions. Students play the role of dispatchers, telling cab- 
bies how to get from one location to another. "Just go three streets 
north and two avenues west" would be such a direction. The efficiency 
of the instructions and the profit of the cab company depends on 
many factors such as one-way streets, accidents, and traffic jams. One 
can easily imagine a board game that would model realistic city traffic 
and get students used to the idea of a two-coordinate instruction set. 

The surface of the earth is another familiar example of a two-dimen- 
sional object. Even though it exists in three-dimensional space, we need 
only two numbers, latitude and longitude, to specify any location. A 
dispatcher of ships can give instructions to go 10 miles due east and then 
5 miles due north. On the surface of the earth but not on a flat plane 
the order of these operations makes a difference: going 5 miles due north 
and then 10 miles due east can put a ship at a different position! The 
extent of this difference is an intrinsic indicator of curvature. 

In teaching geometry we should not ignore the interactive video game. 
Today's students take for granted the fact that we can manipulate im- 
ages on a two-dimensional screen by pushing buttons, turning dials, or 
twisting joysticks. Programs like LOGO offer students experience in 
giving simple geometric instructions to move points and objects around 
on a screen. This gives mathematics teachers a chance to introduce any 
number of important concepts, including repeated operations to form 
regular or star polygons and recursive processes for drawing fractal ob- 
jects or space-filling curves. 



Many video games employ wraparound, which introduces interesting 
ideas in different two-dimensional geometries. Frequently when a point 
is guided off the left side of a computer screen, it appears at the same 
height on the right side. This is analogous to the phenomenon on the 
digital radio dial, which just as well might be thought of as operating 
on a circle. A segment with its endpoints identified can be treated as 
a circle. Analogously, if we think of the points on the left side of a 
computer screen as identified with the corresponding points on the right 
side, then we are dealing not with a flat rectangle but rather with a 

But even more can happen. It is often the case that when a point 
moves off the top of the screen, it reappears at the corresponding po- 
sition on the bottom, so we get a cylinder with its top and bottom 
identified. This gives a figure like an inner tube, which mathematicians 
call a torus. The geometry of a torus is in some ways like that of the 
plane, but in other ways it is very different. In the plane any polygon 
that does not intersect itself divides the plane into two pieces. But if we 
take a closed polygon that goes around the top of the torus, it does not 
separate the torus into two pieces: its inside is the same as its outside. 
Related to this phenomenon is the fact that on a torus we can find two 
closed curves that cross at exactly one point (Figure 31), whereas if two 
closed curves in the plane cross (not just touch), they must intersect in 

FIGURE 31. A torus, the mathematical name for a doughnut- 
shaped surface, is a two-dimensional surface in which two closed 
curves can intersect in just one point and in which a closed curve 
need not separate its inside from its outside. 


an even number of points. An unusual object in many ways, the torus 
is ideal for keeping track of pairs of numbers from circles. 

Three-Dimensional Space 

It is a short step from two to three dimensions. From the two- 
dimensional village layout, we can move to the model of a city, where 
we have a height for each location as well as a position on the grid. 
We can augment taxicab geometry with elevator geometry. We specify 
a position by three numbers, for example, E3N4U9, referring to the 
ninth floor of a building at location E3N4. We can then determine 
an algorithm for getting from this location to E7N2U5. Note that in 
this particular geometry it makes a big difference in what directions 
one moves. The usual algorithm would be D9E4S2U5. Beginning with 
D4 gets you to the right level but in the wrong building! The situation 
would be different for a game played on a jungle gym, with instructions 
to move from one position to another by going a certain distance left 
or right, forward or back, up or down. In this case we can carry out the 
instructions in any order. 

Another three-dimensional geometry arises if we want to specify the 
position of an airplane, giving its longitude, latitude, and altitude. Once 
again, it makes a difference in which order we give the numbers that 
indicate a given location or the directions for getting from one point to 

Higher-Dimensional Spaces 

The intuitions that students accumulate in dealing with coordinate 
pairs in the plane and coordinate triples in three-dimensional space lead 
naturally to coordinate geometry in higher dimensions. A thorough un- 
derstanding of two and three dimensions provides an important foun- 
dation for the powerful generalizations of vector and matrix algebra in 
science and engineering, in economics and social science, and especially 
in computer science and graphics. We illustrate this progression with 
two examples. 

The vertices of a square can be given by four points (0,0), (1,0), (1,1), 
and (0,1). To obtain the vertices of a cube, we can take the points of a 
square with zero in the third coordinate and then move the square one 
unit in the third direction to obtain four more vertices, with a 1 in the 
last coordinate: 

(0,0,0), (1,0,0), (1,1,0), (0,1,0), 
(0,0,1), (1,0,1), (1,1,1), (0,1,1). 



FIGURE 32. Generalizing the Py- 
thagorean theorem to three dimensions 
by applying it to two different triangles 
found in a rectangular box. 

Thus we can describe either the square or the cube as having vertices 
that are either or 1 in each coordinate. 

The procedure generalizes automatically: to obtain the vertices of a 
hypercube, we start with the eight vertices of a cube and put in the 
final coordinate and then "move the cube in a fourth direction" to obtain 
eight more points with 1 the last coordinate: 

(0,0,0,0), (1,0,0,0), (1,1,0,0), (0,1,0,0), 
(0,0,1,0), (1,0,1,0), (1,1,1,0), (0,1,1,0), 
(0,0,0,1), (1,0,0,1), (1,1,0,1), (0,1,0,1), 

(0,0,1,1), (1,0,1,1), (1,1,1,1), (0,1,1,1). 

We thus obtain the sixteen vertices of a hypercube, with or 1 in each 
of four coordinates. It is this sort of representation that is ideal for 
communicating with a computer. 

A second topic that generalizes in a very nice way is the Pythagorean 
theorem. If we think of this theorem as a way of calculating the length of 
the diagonal of a rectangle with given sides, then the extension to three 
dimensions is immediate: given a solid bounded by rectangular sides, 
we first apply the theorem to one side and then apply it to a rectangle 
built over the first diagonal (Figure 32). We easily get e 2 = c 2 + d 2 = 
c 2 + (a 2 + b 2 ), so the length of the diagonal of a rectangular prism with 
sides a, b, and c is Va 2 + b 2 -f c 2 . The pattern is established, and the 
distance formula in four-dimensional space follows almost immediately. 
Students can then calculate the lengths of diagonals of the hypercube 
with the 0-1 coordinates. It turns out that the length of the major 


diagonal of a four-dimensional cube say from (0,0,0,0) to (1,1,1,1) 

is \/4 = 2, which is twice the length of a side. 


The coordinate descriptions that are so useful in giving locations and 
direction in familiar spaces of one, two, and three dimensions work 
equally well for phenomena whose specification requires more than 
three numbers. Exploratory data analysis, a statistical technique for 
dealing with these representations, is one of the most important appli- 
cations of dimensions in current research. The ability to visualize and 
interpret multidimensional data sets may be one of the best gifts we can 
present our students in this modern age. 

Some of the most useful and interesting examples of higher-dimension- 
al phenomena occur as configuration spaces collections of geometric 
objects representing certain structures or motions in the natural world. 
The most familiar spaces are the one-dimensional collection of points 
on a line, the two-dimensional collection of points in a plane, and the 
three-dimensional collection of points in space. But we can also consider 
the collection of lines in the plane, the collection of planes in space, the 
collection of all possible circles in a plane, or the collection of spheres 
in space. We illustrate this process by presenting several examples of 
phenomena that lead to higher-dimensional configuration spaces. 

Consider the following (slightly unrealistic) situation: The lighting 
director of our local theater has to arrange a set of lights over the stage so 
as to illuminate certain parts of the floor at certain times. Sometimes the 
size of a spot is supposed to change during the course of a performance. 
Sometimes one colored circle is supposed to be contained in another. 
How can she keep track of all the circles of light and then design lighting 
directions so that an assistant can carry them out? 

In this particular theater the lights all have the same form. A single 
bulb is suspended from a wire hanging down from the ceiling, and a 
conical shade directs the light out in a beam that meets the floor in 
a disc of light. The sides of the shade come down at a 45 angle, so 
the radius of the disc is equal to the height of the bulb above the floor 
(Figure 33). This makes it easy for the director to specify the location 
of any light, since she can indicate the position of the center of the disc 
using the same coordinates that the director of the play uses to give 
her instructions. That uses two coordinates, but the lighting director 
needs another number to represent the radius of the disc. She could, 
as an alternative, specify the height of the bulb above the floor, since 
in this idealized situation these two numbers are the same. Hence any 
particular disc can be represented by three coordinates, the first two 



FIGURE 33. A spotlight with a shade set at a 45 angle will 
illuminate a spot on the floor of a stage whose radius equals the 
height of the light above the stage. 

being the location of the center and the third giving the radius (or, in 
our special case, the height). 

In this way we see that the collection of discs in the plane is three- 
dimensional; this collection is an example of a configuration space, each 
disc representing one element in the configuration of spotlights. To 
exploit the three-dimensionality as a bookkeeping device, the director 
can record the position of each light by giving three coordinates: for 
example, (6, 8, 5) refers to the light with center at the (6, 8) position on 
the floor and a radius (or height) of 5. 

To call this a space indicates something more than convenience of 
recording. It is a signal that the arithmetic of the coordinates reflects 
properties of the geometry of lights. For example, a spotlight with coor- 
dinates (6, 8, 5) stays on the stage, while the light (6, 4, 5) shines off the 
front of the stage. It is easy to determine a rule to tell when a light stays 
away from the front rim of the stage, namely that the second coordinate 
be larger than the third. 

More complex problems facing the lighting director can also be solved 
by referring to the coordinates. For example, when will one spot be en- 
tirely separate from another? In words, this happens when the distance 
between the points in the plane given by the first two coordinates is 
greater than the sum of the third coordinates. In symbols, the condi- 
tion is expressed by ^(jc - x') 2 + (y -y') 2 > r + r' . 

In this configuration space the three coordinates do not play the same 
sorts of roles; so even though the geometry of the configuration space is 
three-dimensional, it treats the last coordinate differently from the first 
two. It is not identical to the usual geometry of ordinary three-space, 
where the Pythagorean theorem treats all coordinates the same way. An 
important aspect of configuration spaces are the special symmetries they 


The Fourth Dimension 

Sooner or later everyone hears that time is the fourth dimension. 
That idea, however, limits the idea of dimensionality. Already in the 
last century writers realized that there are many situations in which time 
can be viewed as a fourth dimension, but by no means does it demand 
any special role as the fourth dimension. When physicists, especially 
relativity physicists, specify an event by giving three space coordinates 
and one time coordinate, they are using a four-dimensional configura- 
tion space. This space has its own geometry that is not the same as the 
geometry of four-dimensional Euclidean space, where distance is given 
by the generalized Pythagorean theorem. In the theory of relativity the 
distance between two events is given by the expression 

- *') 2 + (y- y'Y + (2 - z') 2 - (/ - *') 2 , 

where time is measured in special units related to the speed of light. 

The three-dimensional configuration space of spotlights provides a 
useful analogy for a four-dimensional space used in molecular modeling. 
The atoms that make up a molecule can be represented by small spheres 
of different radii. The description of a particular molecule, like the 
description of stage lighting, consists of a list of spheres of different sizes 
in different positions. Each sphere requires three coordinates to specify 
its center and one coordinate for the radius. Thus the configuration 
space of atoms is four-dimensional, and a molecule is a collection of 
such atoms arranged in a particular formation. 

Using the language of the configuration space, we can describe a 
molecule to a computer and ask it to display different views. If we 
ask the computer to check that two atoms do not intersect, this involves 
an algebraic condition in four coordinates, namely 

(x - Jc') 2 + (j> - /) 2 + (* - z') 2 - (r + r') 2 > 0. 

The geometry of this configuration space is much closer to that of rela- 
tivity theory than it is to ordinary Euclidean four-dimensional geometry. 
Interestingly it is this sort of question avoiding intersections that ap- 
pears in the science of robotics, using large numbers of coordinates to 
keep track of objects moving through configuration spaces of high di- 

Suppose each light on our sample stage possesses a rheostat that can 
control the current hence the brightness of the spot. If we add 
brightness to the coordinates of the spotlight, then the configuration 


space will be four-dimensional. If we want to encode the color of each 
spotlight as well, then the dimensionality jumps again. The specification 
of color requires three more coordinates representing either hue, satu- 
ration, and value or the relative amounts of red, yellow, and blue (for 
pigments) or red, green, and blue (for lights). So the lighting director 
will now have seven coordinates for each spotlight: two for floor posi- 
tion, one for radius, one for brightness, and three for color. Thus even a 
simple example can lead to a configuration space of high dimensionality. 
Relativity physics began by considering four-dimensional collections, 
with three dimensions for space and one for time. Recently modern 
physics has become much more complicated. Some current models keep 
track of seven dimensions that act like space and four that act like 
time, to give an 1 1 -dimensional configuration space. Another important 
model uses a configuration space with 26 dimensions. In each case the 
choice of the model depends to some degree on the kinds of mathematics 
that apply in these dimensions, as an aid to keeping track of the complex 
interrelationships among events in these high-dimensional spaces. 

Statics and Dynamics 

Here's yet another type of configuration space, set up by a simple 
story. For the school sculpture show two students want to decorate the 
back wall of the hall with a pattern of plastic strings. They decide to 
stretch them from the left-hand edge of the wall down to the floor. By 
trial and error the week before the show, they come up with a pleasing 
design, using more than twenty strings. They can't leave them up until 
the show so they have to find a way of recording the positions so they 
can put them up again later. How many numbers do they need to specify 
the position of each string? What is the dimensionality of the collection 
of strings? 

It is easy to see that the dimensionality of this configuration space 
is two: it takes, just two marks to locate a given string, one along the 
floor and one up the left edge of the wall, and each of these locations 
can be specified by a single number. The pair of numbers (4, 3), for 
example, could represent the string that goes from the point four feet 
over on the floor to the point three feet up on the wall edge (Figure 34). 
The collection of pairs, one pair for each string, tells the positions of 
all strings. It is even possible to record these ordered pairs in a specific 
sequence so the students will know which order to follow when they 
replace them. 

In a way this coding is like the old game of "connect the dots" where a 
polygon is determined by a sequence of ordered pairs, so by connecting 


FIGURE 34. A configuration space of 
two dimensions can represent the positions 
of strings that run from the floor to loca- 
tions on the left edge of the wall. 

the dots in order we draw the polygon. In our sculpture story the basic 
elements are not points but segments: by forming the sequence of string 
segments, we re-create the wall sculpture. 

If we increase the dimensionality of the configuration space, we can 
allow the bottom of the string to be placed anywhere on the floor, with 
the top still somewhere on the left edge of the wall. We still need one 
number for the height, but now the record will have to include two num- 
bers for the floor coordinates. The collection of segments would then 
be three-dimensional, yielding greater possibilities of more interesting 

By allowing the strings to start anywhere on the vertical wall and 
end up anywhere on the floor, we would have a realization of a four- 
dimensional system. Simple algebra would then enable one to predict, 
for example, whether or not two strings are going to intersect. When we 
are laying strings along a wall, it is commonplace for them to intersect. 
Such intersections are rare if we are in a three-dimensional collection 
and rarer still for the four-dimensional system of segments in space. 
It is also interesting to look for configurations of segments that cor- 
respond to familiar configurations in ordinary space. What collection 
of segments in a two-dimensional configuration space corresponds to a 
line joining two points? What segments in a three-dimensional collec- 
tion correspond to a coordinate plane in three-space? Questions such 
as these can yield striking and unpredictable visual effects in the string 

The dimensionality of a configuration space becomes especially im- 
portant when we consider dynamic problems. When a point is moving 
on a line, we can describe its state at any given time by giving two num- 
bers, one for its position and a second for its velocity. The state space 
is therefore two-dimensional, and a point moving according to a given 
physical law, like a ball bobbing up and down on a spring, will describe 
a curve in that state space. Similarly a point moving in a circle, like a 



swinging pendulum, will have a two-dimensional state space giving its 
angular position and angular velocity. 

The state space of a point moving in a plane will be four-dimensional, 
with two points for location and another two for velocity. Scientists 
analyzing the motion of a satellite have to work in a six-dimensional 
state space, with three coordinates for position and three for velocity. 
The laws of physics will restrict the actual states of a system to some 
lower-dimensional space. Indeed, scientists devote a good deal of effort 
to analyzing the shapes of these spaces. For example, the motion of two 
pendulums corresponds to a curve on a torus in four-dimensional space. 
The study of such high-dimensional dynamical systems is an extremely 
important subject in modern applied mathematics. 


When Froebel presented his geometric gifts, he did not want them to 
appear static. One of the first gifts was a display of three basic forms 
suspended by strings in various ways (Figure 35). As the objects rotated, 

FIGURE 35. FroebeFs kindergarten included basic shapes that 
could be hung from eyelets at different positions, then viewed from 
different perspectives to see various cross-sectional shapes. 


FIGURE 36. The central diagonal cross 
section of a cube turns out to be a regular 
hexagon whose six edges cut off triangles on 
each of the six faces of the cube. 

children could observe them from different views and ultimately come 
to an appreciation of their symmetries and structures. 

In the model devised by Froebel, the sphere, the cylinder, and the cube 
all had eyelets attached so that they could be suspended in different 
ways. Because of its symmetry, the sphere had only one eyelet. The 
cylinder had three: one in the center of an end disc, one in the center 
of a side, and one on the rim. The cube also had three: one in the 
center of a face, one in the center of an edge, and one at a vertex. 

The various views of these rotating objects lead to one of the most 
intriguing exercises in understanding forms in space, namely the deter- 
mination of cross-sectional slices. One way to visualize this without 
actually applying a knife to a real model is to imagine what would hap- 
pen if we gradually submerged the block in water. How will the shape 
of the water level change? 

The exercise that is most difficult for students is to visualize the shape 
of the "equator" of a cube suspended from a vertex. A student who 
has looked carefully at a real cube will have a much better chance of 
figuring out that the answer is a hexagon (Figure 36). This fact can 
be demonstrated nicely by stretching a rubber band around a cube. A 
cardboard model for the pieces of this decomposition of a cube can be 
made by cutting corners from three squares and placing them on the 
sides of a regular hexagon (Figure 37). 

A transparent plastic cube half filled with a colored fluid can be ma- 
nipulated to show the various slices through the center. If the cube is 
exactly half full, the shape of the liquid's surface will always be a central 
slice that is, a slice through the center regardless of the cube's ori- 
entation. It is a good challenge to then ask students to figure out which 
position of the cube produces the central slice with the greatest area. (It 
is not the hexagonal slice!) 

Already in the last century when Milton Bradley took up the man- 
ufacture of Froebel' s kindergarten materials in the United States, he 



FIGURE 37. By folding this template into a solid figure, one 
gets half of a cube sliced on the central diagonal. Two such solids 
can be reassembled to form the cube by placing the hexagon faces 

included in one of his sets another figure a cone. The conic sections 
are phenomena that can be seen and appreciated long before students 
are introduced to analytic geometry. Once again, a transparent cone 
partially filled with liquid can illustrate the changing conic sections as 
the object rotates. 

FIGURE 38. As the central slice of a six-sided cube yields a 
regular six-sided polygon, so the central slice of a four-sided 
tetrahedron yields a regular four-sided polygon that is, a square. 
The template on the right provides the means for constructing 
half of a tetrahedron; two such pieces make an excellent geometric 


. . FIGURE 39. Appearances can be deceiving: the di- 

\ / rection of the arrowheads changes the apparent length 

/ \ of the lines without changing their actual length. 

The investigation of slices of polyhedral objects leads to an interesting 
puzzle. If we slice a triangular pyramid by a plane parallel to one of 
its faces, we get a series of triangles. If we slice by planes parallel to 
one of the edges, we get rectangles, and in the central position, a square 
(Figure 38). Students can make cardboard polyhedral models of the 
two pieces of this decomposition by cutting and folding an appropriate 
pattern. Many people find it very difficult to put these two identical 
pieces together to form a triangular pyramid. The difficulty seems to 
be a three-dimensional analogue of the optical illusion that makes two 
lines of equal length seem different if we put arrows on the ends (Figure 

Visitors from Higher Dimensions 

Over one-hundred years ago Edwin Abbott Abbott used slicing to il- 
lustrate the dimensional analogy in his classic satire Flatland. 1 It is a 
great exercise to try to take on the viewpoint of A Square, living in a 
two-dimensional universe, especially when he is visited by a sphere from 
a higher dimension. The frustrated attempts of the sphere to teach A 
Square about the third dimension give wonderful insights into the chal- 
lenges of communication and visualization in geometry. (Early parts of 
Flatland may be difficult for some students, and some of the social satire 
may be skipped over at first reading. Abbott was an active education 
reformer and worker for equality who was satirizing the narrow-minded 
attitudes of Victorian England with respect to class society and partic- 
ularly with respect to women. Only at the end does A Square begin to 
gain a more enlightened view of his society.) 

What would happen if we were visited by a sphere from a dimension 
higher than our own? Instead of growing and changing circles in a 
plane, we would see growing and changing spheres in space. We would 
be inclined to interpret such an event as the inflation and deflation of a 
balloon, but the point of the exercise is that such a phenomenon could 
be interpreted equally well as the slices of a hypersphere penetrating our 
three-dimensional universe. 

If A Square were visited by a cube from the third dimension, he 
would see a variety of polygons, depending on the position of the cube 


as it passed different water levels. What would be the analogous three- 
dimensional slices of a four-dimensional hypercube? This is one place 
where computer graphics can be of great help (as in the film The Hy- 
percube: Projections and Slicing). 3 

Slicing techniques are important in many modern scientific appli- 
cations, especially since the development of computer graphics. X- 
ray tomography uses computer graphics in the reconstruction of three- 
dimensional objects from planar sections. Topographers and geologists 
construct and analyze contour maps showing the elevations of different 
configurations above and below the surface of the earth. Similar slicing 
methods are used by biologists, while researchers in materials science 
use computer graphics to show the parts of a three-dimensional surface 
with a given temperature or density. Exploratory data analysis uses 
techniques of projections and slicing to investigate high-dimensional 
data sets from social sciences as well as from the physical and biologi- 
cal sciences. 

Students of calculus will appreciate the power of slicing techniques 
for example, in relating the volume of a surface of revolution to the 
changing areas of its circular cross sections or in finding the contour 
lines on the surface of a graph in three-space. Long before students 
are introduced to the notions of critical point theory, they can already 
understand and appreciate slicing phenomena that relate different di- 
mensions. What happens if we slice a doughnut or a bagel in different 
directions? It is easy to carry out the actual experiments and see that 
there are positions where the slice yields a pair of circles. Less obvious 
is the slice that consists of two interlocked circles. Again, a good way 
to see this would be to experiment with a transparent inner tube filled 
halfway with colored liquid. Geometry can be a surprising observational 


Many combinatorial and algebraic questions arise in the investigation 
of geometric figures; these can be introduced at different educational 
levels, right up to the frontiers of research. How many edges does a 
triangular pyramid have? We can follow FroebePs suggestion and make 
a model out of toothpicks and peas, then count the edges. Or we can 
simply draw a picture of the object (Figure 40) and count the six edges. 

The procedure for drawing such a diagram suggests an algorithm for 
determining the number of edges. Start with a point, then choose a 
distinct point and draw the one edge connecting it to the one we already 
had. Now choose a new point and connect it to the previous two points 
to get two more, for a total of three. (We have to be careful not to 



FIGURE 40. The tetrahedron the simplest regular poly- 
hedra has four triangular faces, six edges, and four vertices. 

FIGURE 41. By adding one new point 
with line-segment connections to each pre- 
vious vertex, one can construct in sequence 
the complete graphs on 1, 2, 3, 4, 5, 6, 
. . . points. 

choose the new point on the line containing a previous edge.) Next 
choose a new point not lying on any of the three lines determined by 
the edges already constructed, and then connect this new point to the 
previous three. This yields three new edges, for a total of six. 

We can repeat this process to draw the figure called a complete 
graph determined by five points (Figure 41). First choose a point not 
on any of the six lines containing previously constructed edges, and then 
connect it to the previous four points to obtain four new edges for a 
total of 10. A similar construction can produce the complete graph on 
six points and more if so desired. 

What is the pattern that emerges from this procedure? It becomes 
apparent if we arrange the results in a table: 

Number of points: 1234 56 
Number of edges: 1 3 6 10 15 

In each case the number of edges is the number of pairs of points, which 
leads directly to the study of combinations. Based on the sequence of 
construction, it is easy to see that the number of edges at stage n is 
the sum of all numbers less than n. For example, the number of edges 
formed by six points is l-f2 + 3-f4 + 5= 15. Some students may know 
the formula n(n -f l)/2 for the sum of the first n integers, perhaps in 
conjunction with the famous story of the young Gauss who used this 
formula to add up all the numbers from 1 to 100. Another type of 
pattern is revealed by the table that the number of edges at any stage 
is the total of the previous number of edges and the previous number 
of vertices. 



FIGURE 42. A display of different triangles determined by com- 
plete graphs shows that every subset of three vertices determines a 
triangle. Hence counting triangles is equivalent to counting triples 
of vertices. 

Counting Triangles 

Spatial perception tests often ask students to extract a simple figure 
from a complicated one. Counting edges is one of the simplest of such 
tasks. Next in difficulty would be counting the number of distinct tri- 
angles (Figure 42). By marking each triangle we can extend our table to 
include the new information: 

Number of points: 1234 56 
Number of edges: 1 3 6 10 15 
Number of triangles: 1 4 10 ? 

To fill in the missing value we can reason from patterns, many of 
which are just like those that relate edges to points. Since there are as 
many triangles as there are distinct triples of vertices, the total number 
of triangles is just the combinations of a certain number of objects 
taken three at a time. Alternatively, as before, we can use a recursion 
relationship: the number of triangles at any stage is the sum of the 
previous number of triangles and the previous number of edges. The 
latter is the easiest to calculate: it shows that the number of triangles 
that can be formed from 6 points is 20. [In general the number for n 
points is n(n - \)(n 2)/6.] 

Students who have studied some algebra will be able to relate these 
numbers to the binomial coefficients: 


(a + b) =a + b 

(a + b) 2 = a 2 + lab + b 2 

(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 

(a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b* 

(a + b} 5 = a* + 5a*b + I0a 3 b 2 + lQa 2 b 3 + 5ab 4 + b 5 

(a + b) 6 = a 6 + 6a*b + 1 5a*b 2 + 20a 3 b 3 + 1 5a 2 b 4 + 6ab 5 + b 6 

Removing the literal factors leaves a shifted version of Pascal's trian- 

1 1 

1 2 


1 3 



1 4 




1 5 





1 6 






The fourth row, for example, gives in succession for n = 0, 1, 2, 3, and 
4 the numbers of objects with n vertices formed from the four points: 
dots, lines, triangles in the middle, with the empty set and the whole set 
at the ends (where n = and n = 4). 

Observant students may see another important pattern that the sum 
of any row is a power of 2. There is a sophisticated way of stating 
this observation: the sum of the numbers of simplices of different di- 
mensions in an /z-simplex including the whole object and the empty 
simplex is 2 n+l . This same relationship can be observed by setting 
both a = 1 and b = 1 in the table of binomial expansions or by relating 
the binomial coefficients to the combinations of n -f 1 elements taken 
k -f 1 at a time. The total number of possible combinations is then 
2" +1 , the total number of subsets chosen from among n -f 1 elements. 
This basic counting argument can motivate many topics in elementary 

Counting Squares and Cubes 

Similar observations emerge if students investigate the numbers of 
vertices, edges, and faces of cubes and hypercubes in various dimen- 
sions. Just as there is a hierarchy of subsimplices within each simplex, 
there is an analogous sequence of squares and cubes within each n- 
dimensional cube. A 3-cube has 8 vertices, 12 edges, and 6 squares, as 
can be verified by an actual count. A square, or 2-cube, has 4 vertices, 
4 edges, and 1 square. A 1-cube is a segment with 2 vertices and 1 edge, 



FIGURE 43. Framework for hypercube: 
two cubes with joined edges yield 16 
vertices and 32 edges. 

FIGURE 44. Shading helps identify two 
horizontal groups of four parallel squares 
in the hypercube. There are six such 
groups in all, three associated with the 
original cube and its displaced copy and 
three associated with the edges that join 
the two cubes. 

and a 0-cube is point with 1 vertex. This data can form the beginning 
of another table: 

DIMENSION: 0-cubes 1 -cubes 2-cubes 3-cubes 4-cubes 
(points) (lines) (squares) (cubes) (hypercubes) 
Vertices Edges Faces Cubes 4-Cubes 









When we try to fill in the missing numbers for a hypercube, the process 
becomes a bit more difficult. We know how to generate a hypercube 
move an ordinary cube in a direction perpendicular to itself. As the 
cube moves, the 8 vertices trace out 8 parallel edges. This yields 12 
edges on the original cube, 12 on the displaced cube, and 8 new edges 
traced by the movement for a total of 32 edges on the hypercube (Figure 

Counting squares presents more of a problem, but a version of the 
same method can be used to solve it. First observe that there are 6 
squares on the original cube and 6 on the displaced one. To these 12 
we must add the srmares trar,ftH rmt hv thft ftriees nf thp mnvina 



FIGURE 45. A group of four vertical 
squares in a hypercube determined by the 
horizontal displacement of the original 
cube. These squares are easier to see 
when background lines are removed, as 
in the lower figure. 

It helps to group edges and squares in parallel bundles. The edges in 
the hypercube come in four groups of 8 parallel edges. Similarly the 
squares can be classified in four groups of 4 parallel squares, one such 
square through each vertex. Two horizontal groups are rather easy to see 
(Figure 44); another group of four vertical faces become clearer when 
we remove some of the extraneous lines (Figure 45). 

Student teams can easily identify the remaining three groups of four 
squares. It is easier to do this when the four squares do not overlap 
and relatively more difficult when the overlap is large. The entire set 
consists of 24 squares. 

Grouping edges or faces is particularly effective when an object pos- 
sesses a great deal of symmetry, as does the hypercube. We can study the 
relation between symmetry and grouping by looking at different dimen- 
sions. Symmetries of a cube, a square, or a segment arise by permuting 
the edges at each vertex in different ways and by moving each vertex 
to another position. The collection of all symmetries of the cube or 
hypercube is an important example of a group, an algebraic structure 
that reflects geometric properties. The symmetry group of a cube is 
the collection of permutations of its vertices that preserve its structure. 
The attempt to codify the relation of permutations to symmetries of al- 
gebraic and geometric structures provided considerable impetus for the 
development of modern algebra during the past two centuries. Even now 
symmetry groups continue to fuel theoretical work in atomic physics. 



The crucial observation about the hypercube is that it is so highly 
symmetric that every point looks like every other point: if we know 
what happens at one vertex, we know what happens at all vertices. For 
example, at each of the 16 vertices of the hypercube there are 4 edges, 
for a total of 64. But this process counts each edge twice, so the actual 
number of edges is half of 64, or 32. 

At each vertex there are a certain number of square faces. How many? 
As many as there are ways to choose two edges from among the four 
edges that meet at the vertex. Once we have chosen one edge from 
among the four, there remain three possibilities for the second; together, 
these yield 12 pairs. As before, each pair of edges appears twice in this 
list, once in each order. So these 12 pairs yield 6 different squares at 
each vertex. All 16 vertices together then yield 96 squares. But each 
square is counted four times, once for each of its vertices. Hence the 
true total is 96/4 = 24 squares in a hypercube. This reasoning confirms 
the direct count of six groups of four squares that we saw in drawings 
of the hypercube, but it is reached by a method that would work even 
if applied to a five-dimensional cube. 

Seeking Patterns 

Advanced students can express these results in a general formula. 
Let a (k, ri) denote the number of fc-cubes in an -cube. To calculate 
n (k, n) we begin, as before, by counting how many fc-cubes there are at 
each vertex. Each /c-cube is determined by a subset of k distinct edges 
from among the n edges emanating from each vertex. Therefore the 
number of /c-cubes at each vertex is C(/c, ri) = () = n\/k\(n - k)l, the 
combination of n things taken k at a time. Since there are C(k, n) k- 
cubes at each of the 2 n vertices, the total number of /c-cubes appears to 
be 2 n C(k,ri). But in this count each /c-cube is counted 2 k times, so we 
divide by that number to get the final formula: n(fc,) = 2 n ~ k C(k,n). 

Remembering the pattern of powers of 2 that come from the sums of 
rows in the simplex table, we naturally seek a similar pattern for cubes. 
In this case the entries in each row add up to a power of 3: 

DIMENSION: 0-cubes 1 -cubes 2-cubes 3-cubes 4-cubes 

(points) (lines) (squares) (cubes) (hypercubes) 
Vertices Edges Faces Cubes 4-Cubes Sum 




























FIGURE 46. Subdivision of the sides of segments, squares, and 
cubes (and even hypercubes) into three equal parts yields 3, 9, 27, 
or 81 similar small objects always a power of 3. 


There are several ways to react to this observation. We can generate an 
additional row of the table to gain some additional information, but the 
conjecture is fairly firmly established with the five completed rows. We 
can observe that each entry is the sum of twice the entry directly above 
it plus the entry to the left of that one, so the sum of entries in one row 
is three times the sum of entries in the previous row an argument that 
can easily be translated into a formal proof by mathematical induction. 
We may also use the explicit formula for the number of fc-cubes in an 
rc-cube, to sum a typical row: 

n (0, n) + n (1, n) + +Q (n - l,n)+ a (n,n) 
= 2" + C(l, n} 2n ~ l + C(2, n}T~ 2 + - - - 4- C(n - 1, n)2 + C(/i, n) 

All these approaches help explain why the rows sum to power of 3. 
But perhaps the most satisfying observation that justifies this fact is 
that we may divide the sides of an -cube into three equal parts whose 
projections divide the entire cube into 3" small cubes (Figure 46). The 
result is a small cube coming from each vertex of the original cube, 
one from each edge, one from each two-dimensional face, and so on. 
The final small cube is in the center. Thus the total number of small 
-cubes, which is 3", is equal to the sum of the number of A: -cubes in 
the rc-cube since there is one small n-cube for each point, edge, face, 
3-cube, etc. 

One of Friedrich FroebeFs kindergarten gifts was a cube subdivided 
into 27 small cubes. He would have liked this final demonstration. 


1. Abbott, Edwin Abbott. Flatland. London, England: Seeley & Co., 1884; numerous 
reprintings, especially BlackwelFs (1926) and Dover (1952). 

2. Banchoff, Thomas. Beyond the Third Dimension: Geometry, Computer Graphics, and 
Higher Dimensions. New York, NY: Scientific American Library, W. H. Freeman & 
Co., 1990. 

3. Banchoff, Thomas and Strauss, Charles. The Hypercube: Projections and Slicing. 
Chicago, IL: International Film Bureau, 1978. 

4. Botermans, Jack. Paper Capers. New York, NY: Henry Holt and Company, 1986. 

5. Barnsley, Michael. Fractals Everywhere. San Diego, CA: Academic Press, 1988. 

6. Critchlow, Keith. Order in Space. New York, NY: Thames and Hudson, 1969. 

7. Davidson, Patricia and Wilicull, Robert. Spatial Problem Solving with Paper Folding 
and Cutting. New Rochelle, NY: Cuisenaire Company of America, 1 984. 

8. Dewdney, Alexander. The Planiverse. New York, NY: Poseidon Press, 1984. 

9. Ernst, Bruno. Adventures with Impossible Figures. Norfolk, England: Tarquin Publi- 
cations, 1986. 


10. Heinlein, Robert, "...and He Built a Crooked House." In Fadiman, Clifton (Ed.): 
Fantasia Mathematica. New York, NY: Simon & Schuster, 1958. 

1 1 . Froebel, Friedrich. Education by Development. New York, NY: D. Appleton & Com- 
pany, 1899. 

12. Gardner, Martin. Mathematical Carnival. New York, NY: Alfred A. Knopf, 1975. 

13. Hix, Kim. Geo-Dynamics. Conestoga, CA: Crystal Reflections, 1978. 

1 4. L'Engle, Madeleine. A Wrinkle in Time. New York, NY: Farrar, Straus, and Giroux, 

1 5. Manning, Henry Parker. The Fourth Dimension Simply Explained. New York, NY: 
Macmillan and Co., 1911. 

16. Pearce, Peter and Pearce, Susan. Polyhedra Primer. Palo Alto, CA: Dale Seymour 
Publications, 1978. 

1 7. Pearce, Peter. Structure in Nature Is a Strategy for Design. Cambridge, MA: MIT 
Press, 1978. 

18. Peterson, Ivars. The Mathematical Tourist. New York, NY: W.H. Freeman & Co., 

19. Rucker, Rudy. The Fourth Dimension: Toward a Geometry oj 'Higher Reality. Boston, 
MA: Houghton Mifflin, 1984. 

20. Tufte, Edward. The Visual Display of Quantitative Data. Cheshire, CT: Graphics 
Press, 1983. 

2 1 . Wells, David. Hidden Connections, Double Meanings. Cambridge, England: Cam- 
bridge University Press, 1988. 

22. Wiebe, Edward. Paradise of Childhood, Golden Jubilee Edition. Milton Bradley (Ed.), 
including a "Life of Friedrich Froebel" by Henry Blake, Springfield, MA: Milton 
Bradley Company, 1910. 

23. Winter, Mary Jane, et al. Spatial Visualization. Middle Grades Mathematics Project. 
Menlo Park, CA: Addison- Wesley, 1986. 



One of the principal factors in human intellectual development is our 
desire to make sense of the physical and biological worlds in which we 
live. We search historical records for clues that explain our present con- 
dition, and we devise theories that might predict the future. In nearly 
every description of the past or forecast of the future, prominent factors 
include quantitative attributes: length, area, and volume of rivers, land 
masses, and oceans; temperature, humidity, and pressure of our atmo- 
sphere; populations, distributions, and growth rates of species; motions 
of projectiles, tides, and planets; revenues, costs, and profits of eco- 
nomic activity; rhythms, intensity, and frequency of sounds, light, and 

Perceptive observers have noted that patterns in objects can be mod- 
eled by numbers in ways that aid reasoning. It may be an exaggeration 
to say, as Lord Kelvin once claimed: 32 

When you can measure what you are speaking about and express it in numbers, 
you know something about it; but when you cannot measure it, when you cannot 
express it in numbers, your knowledge is of a meager and unsatisfactory kind. 

But it is not an exaggeration to say that the number systems of mathe- 
matics are indispensable tools for making sense of the world in which 
we live. 

The human fascination with numbers is also reflected in countless 
examples of whimsical or superstitious numerology. From the Greek 
Pythagoreans to Martin Gardner's fictional Dr. Matrix, 10 people have 


found meaning both sublime and sinister in numerical values 
attached to letters, words, names, places, and dates. The endless va- 
riety of patterns in numbers has piqued the mathematical curiosity in 
millions of professional and amateur mathematicians of all ages. Un- 
fortunately, those same patterns have served as the basis of various 
pseudoscientific enterprises from astrology to numerology. 


Given the fundamental role of quantitative reasoning in applications 
of mathematics as well as the innate human attraction to numbers, it is 
not surprising that number concepts and skills form the core of school 
mathematics. In the earliest grades all children start on a mathemat- 
ical path designed to develop computational procedures of arithmetic 
together with corresponding conceptual understanding that is required 
to solve quantitative problems and make informed decisions. Children 
learn many ways to describe quantitative data and relationships using 
numerical, graphic, and symbolic representations; to plan arithmetic 
and algebraic operations and to execute those plans using effective pro- 
cedures; and to interpret quantitative information, to draw inferences, 
and to test the conclusions for reasonableness. 

The skills required for these tasks are contained in the arithmetic of 
various number systems and in the generalizations of arithmetic reason- 
ing to elementary algebra. The public recognizes these number systems 
by their common names (whole numbers, fractions, decimals); math- 
ematicians use more formal terms (integers, rationals, real numbers). 
Regardless of their names, these number systems are well-known parts 
of mathematics and have been taught in school for centuries. Experi- 
enced teachers have devised countless clever strategies for developing 
student skill in solving traditional problem types. So it is entirely rea- 
sonable to ask, "What can be new and exciting about teaching quan- 
titative reasoning?" Surprisingly, the answer ought to be, "Just about 

Influence of Technology 

School arithmetic and algebra have always been dominated by the 
goal of training students to manipulate numerical and algebraic sym- 
bols. The purpose of all this manipulation is to answer arithmetic prob- 
lems or solve algebraic equations. The core of elementary and middle 
school mathematics features addition, subtraction, multiplication, and 
division of whole numbers and fractions; the core of secondary school 


FIGURE 1. Hand-held calcula- 
tors can now display graphs of 
all functions ordinarily studied in 
school mathematics. Some can 
even perform most common types 
of symbolic manipulation to sim- 
plify and solve equations. 

mathematics covers similar operations on polynomial, rational, and ex- 
ponential expressions. 

In the past, proficiency with these routine manipulative skills has been 
a prerequisite for effective use of mathematics. However, the emer- 
gence of inexpensive electronic calculators and computers has changed 
that condition forever. It is now about 1 5 years since the technology 
of transistors, printed circuits, and silicon chips first made hand-held 
calculators available on the mass consumer market. Rapid progress in 
electronics has now produced solar-powered scientific calculators that 
perform arithmetic on numbers that can be entered and displayed in 
decimal, common fraction, or exponential form. Many calculators also 
have single-button subroutines for evaluating elementary functions and 
performing common statistical calculations. Programmable calculators 
offer more powerful capabilities, including graphing, symbolic manipu- 
lation, and matrix operations (see Figure 1). Each of these mathemat- 
ical procedures is available in more powerful and sophisticated form 
through programs that run on desktop computers now widely available 
in schools. 

The computational capabilities of machines both existing and 
envisioned suggest some exciting curricular possibilities. Elementary 


school students can now deal with realistic numerical data very large 
and very small numbers in decimal and fractional form without 
prerequisite mastery of the intricate computational algorithms for 
operations on those numbers. Middle school students can deal with 
questions about variables, functions, and relations expressed in algebraic 
language long before they master the rules for manipulating those ex- 
pressions. In the world outside of school, almost everyone relies on cal- 
culators and computers for fast and accurate computation. But school 
curricula have yet to change significantly in response to these new con- 

Calculators and computers are also having a profound effect on the 
nature of mathematics itself. With access to those tools, mathemati- 
cians can search for patterns in much the way that scientists explore re- 
sults from experiments with systematically manipulated variables. The 
experimental mathematician can test special cases on a computer in a 
small fraction of the time required by "paper-and-pencil" algorithms. 
In many cases these calculations could not be done at all by traditional 
means, and the patterns that emerge would never have been seen. The 
experimental data of mathematics can be sorted, analyzed, and dis- 
played graphically to reveal both regularities and variations. The ulti- 
mate standard for verification remains formal proof by reasoning from 
axiomatic foundations. However, calculators and computers have cre- 
ated a new balance between theorem-finding and theorem-proving. 

Use of calculators and computers for mathematical work has also 
led to a dramatic increase in interest in algorithmic methods and re- 
sults. Many of the deepest and most beautiful results of mathematics 
are those that guarantee the existence of numbers with interesting prop- 
erties or solutions to important equations, yet those same theorems and 
their proofs quite often give no clue as to how one might effectively 
construct the promised object. Mathematical contemporaries of Euclid 
could prove that there is no largest prime number and that any natural 
number whatever can be factored uniquely into a product of primes. 
But mathematicians working today still devote great energy to practical 
and theoretical problems posed by the need to construct large primes 
and to find the promised factorizations of large composite numbers. 
The search for effective and efficient algorithms that will guide com- 
puter procedures has become a central aspect of both pure and applied 
mathematical research in our technology-intensive world. 


Influence of Applications 

A second fundamental change affecting school curricula is the ex- 
tension of quantitative methods to nearly every aspect of contemporary 
personal and professional life. Although numbers have always been use- 
ful, their uses have been rather predictable and limited to well-defined 
familiar problems. Today, quantitative literacy requires an ability to 
interpret numbers used to describe random as well as deterministic phe- 
nomena, to reason with complex sets of interrelated variables, and to de- 
vise and interpret critically methods for quantifying phenomena where 
no standard models exist. Examples are all around us: 

U.S. census figures are used to describe our current population 
and to apportion resources to various social programs. How can 
the population and its characteristics best be counted? 

Several hurricanes strike Central and North America each fall. 
How can the "size" of each be measured in the most meaningful 

The consumer price index is used to calculate cost-of-living in- 
creases in Social Security payments and a number of other salary 
scales. How can inflation best be measured? 

Players on football teams in different conferences are often com- 
pared statistically to see who is best, in part to determine fair 
compensation. What data should be used to rank the quarter- 
backs most accurately? 

Banks, credit card companies, and airline and hotel reservation 
systems process billions of financial transactions daily, using na- 
tional communication networks that are protected against errors 
and unauthorized intrusion. How can secure systems be devised 
and used intelligently? 

Each of these problems and many others of similar complexity and 
significance require the ability to organize, manipulate, and interpret 
quantitative information. Skill in traditional written algorithms for 
arithmetic and algebra or in solution of traditional "types" of word 
problems is not only insufficient preparation for those tasks, it is largely 

Quantitatively literate young people need a flexible ability to identify 
critical relations in novel situations, to express these relations in effective 
symbolic form, to use computing tools to process information, and to 
interpret the results of those calculations. The underlying mathematical 
ideas used in this modeling often extend beyond numbers and fractions 
to matrices, linear algebra, and the arithmetic of congruence classes. 
The useful computational tools extend beyond hand-held calculators to 
spreadsheets, data bases, and dynamic simulations. 


Influence of Psychological Research 

Another recent change in conditions for teaching about quantity in 
school mathematics is the emergence of an extensive body of research 
on human cognition. While there is a long history of research on math- 
ematics teaching and learning from a psychological perspective, the past 
thirty years has seen an unprecedented search to identify the ways that 
young people develop understanding of number systems and their ap- 
plication. As a consequence, researchers are acquiring rich insight into 
the interplay between human cognitive development and the concepts, 
principles, and skills that we want young people to learn. This research 
shows real potential for informing decisions about design of curricula 
and instructional approaches in school mathematics. 


The convergence of rapidly escalating demands for social and sci- 
entific application of quantitative skills with powerful new technolo- 
gies that support those skills has prompted reconsideration of goals for 
school mathematics. To paraphrase the title of a 1982 report of the 
Conference Board of the Mathematical Sciences, 29 we are still asking, 
"What quantitative abilities will be fundamental in the future of math- 
ematics?" Despite extensive professional debate over the past decade, 
there is as yet no consensus on a prudent course of change, and most ev- 
idence suggests that schools have not moved toward any radical change. 

In a mature branch of mathematics such as number theory, analysis, 
or algebra, many fundamental concepts and operations can be presented 
in a coherent system of abstract ideas a few definitions and axioms 
from which every other fact and principle follow logically. But this rig- 
orous, efficient organization of contemporary mathematics is only the 
final product of an historical process in which fundamental ideas were 
used informally long before they become formal definitions and theo- 
rems. Furthermore, practical working knowledge requires more than an 
ability to recite or derive formal principles. It requires the ability to 
recognize quantitative relationships in a broad range of concrete situa- 
tions as well as the technical skills to represent and reason about those 

In thinking about school mathematics many mathematicians and 
teachers have argued that the best guide is a curriculum that retraces 
the meandering historical path by which numerical techniques have de- 
veloped. Others suggest that we should capitalize on structural insights 
that have emerged at the end of that path, to provide for children a more 
efficient way to develop number concepts and techniques. There is little 


research evidence to suggest the right choice among these options, but 
it seems safe to say that quantitative understanding requires grasp of 
insights provided by each perspective. It seems important to convey to 
students, as quickly as possible, effective modern techniques for repre- 
senting and reasoning about numerical data. But that instruction will 
undoubtedly be more successful if it is informed by understanding of 
the roots of numerical techniques in human experience and the path by 
which ideas and skills have evolved over time. Students must efficiently 
learn concepts, techniques, structural properties, and uses of the number 
systems 33 but with an honest portrayal of the many informal and halting 
ways that new mathematical ideas and methods actually develop. 

Numbers and Operations 

In searching for a framework of fundamental number concepts to be 
developed in school mathematics, it is helpful to begin with a simple 
question: How are numbers used? In common sources such as daily 
newspapers, cookbooks, instruction manuals, or household budgets, one 
will find a long list of situations in which numbers play a vital role. 
Furthermore, skill in quantitative reasoning is a critical prerequisite for 
success in any scientific, technical, or business occupation, and the list 
of ways that numbers are used in those fields is both long and diverse. 

Designers of curricula are understandably frustrated by the challenge 
of selecting material that will prepare students for all problem-solving 
situations they might reasonably face outside school. However, a search 
for common features in quantitative reasoning tasks shows that they can 
be grouped into a few categories. One common analysis of number uses 
shows that every example involves one of three basic tasks: 

1. MEASURING. To use operations of arithmetic to reason about 
size to answer questions like "How many?" or "How much?" 

2. ORDERING. To use numbers to indicate position in a sequence 
with the relations of "greater than" or "less than." 

3. CODING. To provide identifying labels for objects in a collection. 

Illustrations of these different tasks abound in ordinary life. Here are 
some particular examples: 7 ' 33 

Standard measurement tasks involving concepts such as length, 
area, volume, mass, and time all employ numbers to indicate 
size. The operations of addition, subtraction, multiplication, and 
division correspond directly to operations such as joining, com- 
paring, or partitioning of objects that numbers measure. Other 
important concepts such as velocity, acceleration, and density 


also use numbers to indicate size, but they are usually derived 
by operations on basic number measurements. 

As customers enter a store they are often assigned numbers to 
indicate the order in which they will be served. Customers who 
enter early will have lower numbers than those who enter later 
the order of arrival corresponds to the order of service numbers. 
In this case positive whole numbers are used to indicate order. 
It makes no sense to add or multiply service numbers, although 
subtraction might help to estimate expected waiting time. 

The teams in any athletic league are commonly listed in the order 
of their competitive standing, from first through last. However, 
without further information, those rankings tell little about the 
distance between teams in that order. 

In analyzing games of chance each possible outcome is assigned 
a number between and 1 as its probability. Event A being 
more likely than event B corresponds to the probability p(A) be- 
ing greater than the probability p(B). Furthermore, if A and B 
are disjoint, p(A U B) should equal p(A) +p(B). In this situa- 
tion the assignment indicates a measure of likelihood. But those 
measures are then used to order events by likelihood. The oper- 
ation of union for disjoint events corresponds to the addition of 
rational numbers. 

The uniforms of athletic teams generally have numbers for each 
individual player. While the numbers sometimes indicate an 
assigned position, arithmetic operations or relations involving 
those numbers seldom give any significant information. These 
numbers are used solely as labels. 

This taxonomy of uses of numbers might seem too obvious to men- 
tion. But it offers the first step toward a framework for organizing the 
profusion of quantitative reasoning tasks into manageable families a 
way to find significant themes among the details of number concepts, 
skills, and applications. With suitable refinement the taxonomy can 
help reveal to both teachers and students the experiential root meaning 
of numbers, to focus instruction on the forest as well as the trees. 

For just that purpose Usiskin and Bell 33 have proposed a more de- 
tailed analysis of fundamental kinds of number uses. They suggest six 
different uses of single numbers: 

Counts for discrete collections (populations); 

Measures for continuous quantities (time, length, mass); 

Ratio comparisons (discounts, probabilities, map scales); 

Locations (temperature, time line, test scores); 

Codes (highway, telephone, product model numbers); and 


* Derived formula constants (n in A = rcr 2 ). 

A parallel taxonomy suggests ways that operations on numbers can be 
matched to operations on objects that numbers describe: 

* Addition models putting together or shifting; 

* Subtraction models take-away, comparison, shift, or recovering 
an addend; 

* Multiplication models size change, acting across, or use of a rate 
factor; and 

* Division models ratios, rates, rate division, size change division, 
or recovering a factor. 

While mathematicians and teachers might question the meaning of 
these categories and debate their completeness or independence, it seems 
certain that attention to such analyses will help focus instruction on the 
fundamental task of preparing students to use numbers effectively to 
solve problems. Examples of the different ways that numbers are used 
highlight the essential components in any quantitative reasoning task. 
In simplest form, quantitative reasoning involves phenomena, a num- 
ber system, and a correspondence between phenomena and numbers 
that preserves essential structure. Each object is assigned a number in 
such a way that "similar" objects have "similar" numbers and relations 
among objects corresponding to relations in the number system. To un- 
derstand this modeling process students need extensive experience with 
the structural properties of various kinds of number systems. 

While students must certainly acquire comfortable skill in dealing 
with many specific uses of numbers, they also need to acquire a broader 
perspective on properties that number uses have in common. There 
is clear evidence from research in mathematics education that under- 
standing fundamental structural properties of a mathematical system 
facilitates retention of the system and application to new situations. 
School mathematics should, therefore, emphasize the ways that differ- 
ent types of number systems serve as models of measuring, ordering, 
and coding, together with the ways that standard operations model fun- 
damental actions in quantitative situations. 

Variables and Relations 

Elementary uses of numbers focus on descriptions and inferences con- 
cerning specific quantitative facts the cost of 5 candy bars priced at 
50^ apiece, the area of a field that is 50 feet long and 30 feet wide, or the 
average speed of a car that travels 300 miles in 5 hours. Mastery of con- 
cepts required by such tasks is certainly a central and formidable task 


of school mathematics. However, for quantitative reasoning to yield 
results with greater power than unadorned number facts, it is essential 
that such reasoning be firmly rooted in general patterns of numbers and 
related computations. 

The typical pattern is a relation among two or more varying quantities. 
For example, 

As time passes, the depth of water in a tidal pool increases and 
decreases in a periodic pattern. 

As bank savings rates increase, the interest earned on a fixed 
monthly deposit also increases. 

If a sequence of squares have sides 1, 2, 3, 4, 5, . . . , the areas of 
those squares are 1, 4, 9, 16, 25, 

For any rectangle of base b and height h, the perimeter p is 
2b + 2/z. 

The key mathematical ideas required to reason about such patterns are 
the core concepts of elementary algebra: variables, functions, relations, 
equations, inequalities, and rates of change. In school mathematics 
today students spend a great deal of time working with variables as 
letter names for unknown numbers and with equations or inequalities 
that place conditions on those numbers. Algebra instruction focuses on 
formal procedures for transforming symbolic expressions and solving 
equations to find the hidden value of the variable. 

But those skills are only a small part of the power that algebra pro- 
vides. In each of the examples above, and in countless other similar 
problems, the conceptual heart of the matter is understanding relations 
among several quantities whose values change. The notion of variable 
that students must understand is not simply "a letter standing for a 
number" or "an unknown value in an equation." It must also include 
thinking about variables as measurable quantities that change as the 
situations in which they occur change. 

Variables are not usually significant by themselves, but only in re- 
lation to other variables. In most realistic applications of algebra the 
fundamental reasoning task is not to find a value of x that satisfies one 
particular condition, but to analyze the relation between x and y "for 
all x." The most useful algebraic idea for thinking about relations of 
this sort is the concept of function. 

To develop understanding required for effective application of alge- 
bra, students need to encounter and analyze a wide variety of situations 
structured by relations among variables. They need comfortable un- 
derstanding of relational phrases such as ">> depends on ;c," "y is a 
function of x," or "change in x causes change in y." It is helpful if they 

a Direct Variation 


b Inverse Variation 

c Accelerated Variation 

d Converging Variation 

e Cyclic Variation 

f Stepped Variation 

FIGURE 2. The behavior of fundamental types of relations 
among variables can be seen most readily from typical graphs. 
Graphs (a) and (b) illustrate direct and inverse relations, (c) 
and (d) show accelerated and converging variation, and (e) and 
(f) illustrate cyclic and stepped variation. Virtually all variation 
actually observed is a combination of these basic types. 



develop a repertoire of criteria for characterizing and sorting, by struc- 
ture, the relations they encounter. For instance, the report Science for All 
Americans of the American Association for the Advancement of Science 1 
suggests that students should be sensitive to at least the following kinds 
of relations among variables (see Figure 2): 

Direct and inverse variation as one variable increases, another 
also increases (or decreases) at a similar rate. 

Accelerated variation as one variable increases uniformly, a 
second increases at an increasing rate. 

Converging variation as one variable increases without limit, 
another approaches some limiting value. 

Cyclical variation as one variable increases uniformly, the other 
increases and decreases in some repeating cycle. 

Stepped variation as one variable increases, another changes in 

The idea behind learning properties of whole families of relations is 
typical of all mathematics: recognition of structural similarities in ap- 
parently different situations allows application of successful reasoning 
methods to new problems. With the focus of algebra directed at vari- 
ables and functions, equations and inequalities can be used to represent 
specific conditions: 

If the height of a projectile is a function of its time in flight with 
rule h(t] = - 1 6t 2 + 88r, the equation - I 6t 2 + 88t = asks when 
the projectile is at ground level (see Figure 3). 

If the population of a country (in millions) is a function of time 
with rule/?(0 = 120(2- 03 0, the inequality 120(2- 03 '> < 200 asks 
when the population will stay below 200 million (see Figure 4). 

Of course, thinking about quantitative relations as functions encour- 
ages reasoning that extends beyond familiar equation-based questions 
to notions of rates of change, maxima and minima, and overall trends. 

FIGURE 3. The standard parabolic trajectory be- 
comes visible in a graph of the height of a projectile as 
a function of its time in flight. 



FIGURE 4. The common exponential curve repre- 
sents the equation that describes the growth of a 
country's population. 

While these questions are not generally considered central to school 
algebra, there can be no doubt that they are important considerations 
in any situation that algebraic expressions model. 


The first step in effective problem solving is to analyze the problem 
to identify number concepts that match problem conditions. But that 
is only part of the modeling phase of solving problems the conceptual 
description of what is known. Problem solving also requires inference of 
new information that gives new insight. In mathematics that inference 
invariably relies on systematic techniques for representing and manip- 
ulating information and, in quantitative problems, on procedures for 
calculating results. Recent analyses of mathematics pedagogy describe 
this kind of knowledge as procedural knowledge, in contrast to the con- 
ceptual knowledge required to identify fundamental ideas. 14 Procedural 
knowledge includes techniques required to represent information and to 
execute operations that yield solutions to specific numerical problems. 

Numerical Representation 

Formal mathematics is a subject that deals with mental constructs that 
are abstracted from patterns in objects. But mathematicians have also 
devoted a great deal of energy to find ways of representing ideas in con- 
crete form. Their goal is a system of symbols that convey mathematical 
information effectively in unambiguous and compact form. 

Representation of ideas serves as an aid to memory and as a medium 
for communication. In mathematics the representations become ob- 
jects of study themselves sources of new abstractions that, surprisingly 
often, serve as useful models of unanticipated patterns in concrete sit- 


The fundamental idea that enables efficient representation of num- 
bers is the place value system of numeration. Every whole number has 
a unique representation in the standard base 10 numeration system, 
and rational numbers can be expressed using decimal fractions or as 
quotients of whole numbers. These customary systems are sometimes 
replaced by place value systems with different bases, especially in cases 
where the alternative base has obvious advantages for a particular pur- 

While the place value system is taken for granted today, in thinking 
about mathematics teaching it is worth remembering that the evolution 
of such a powerful representation scheme took a very long time. There 
are signs in the record of early Mesopotamian mathematics that a base 
60 numeration system, using few number symbols, was understood and 
used. However, the place value concept eluded Greek mathematicians 
in their golden era. It was not until Hindu mathematicians of the eighth 
century saw how to use (zero) as a place holder that the foundation 
of place value notation was secured. 

The second major task in representing numerical information is to 
express relationships that are true for all numbers, for many numbers, 
or for certain unknown numbers. The fundamental mathematical con- 
cepts involved are variables, functions, and relations. We now rou- 
tinely use letters to name variables and to write rules for functions and 
relations. But again, it is worth recalling that the historical develop- 
ment of contemporary algebraic notation is a long story testimony to 
the fact that the use of literal variables with algebraic syntax such as 
y = x 3 - 4(x + 2)" 1 is anything but obvious. 

Graphical Representation 

While traditional place value numerals and algebraic expressions are 
the most important symbolic forms for recording quantitative informa- 
tion, many other representational forms are in common use. The most 
popular are those that identify numbers with points in a geometric line 
or pairs of numbers with points in the plane. 

For example, conditions on variables such as \x - 2| < 3 are quite 
common in algebra and its applications. The solutions can be given in a 
similar symbolic form, but it has become almost as common to display 
the results on a number line graph (Figure 5). Although this repre- 
sentation is certainly not as compact or computationally useful as the 
symbolic version, it conveys quickly a total picture of the quantitative 

The use of visual representation to display a relation among quan- 
titative variables is especially effective when one variable is a function 



! I I L 
-4 -3 -2 -1 


S S I J 


FIGURE 5. Intervals portrayed on a number line provide an 
effective picture of the points that satisfy \x ~ 2\ < 3. 

of another. Here's a common example: The position of a piston with 
4-inch stroke in an engine running at 3000 rpm is given by the function 
y = 2sin(1007r)> where t is time measured in seconds. The pattern of 
piston positions is well displayed by a function graph (Figure 6). Like 
the number line graph, this visual image of a relation between two vari- 
ables is not particularly effective as a computational aid, but it does 
convey the significant periodic pattern in piston motion in a way that is 
far less apparent from the symbolic form. 

The use of number lines and coordinate graphs is a very familiar 
mathematical technique. However, the advent of graphing calculators 
and computer software has made a dramatic impact on the ease of 
producing graphs and thus on their usefulness. It is now possible to 
produce graphs quickly and accurately both from formulas and numer- 
ical data drawn from scientific experiments or from large data bases 
that computers have made accessible. As a result, graphic displays are 
becoming common and increasingly sophisticated. Thus it is impor- 
tant for mathematics students to become adept at interpreting graphic 
representations intelligently and to understand the connections among 
symbolic, graphic, and numerical forms of the same ideas. 

There has been great optimism about the potential payoff of using 
these linked multiple representations as an aid in teaching. However, 
early experiments have revealed the fact that the messages provided by 
graphs are not grasped by young learners as easily ais might be expected, 
while the effects of scale and the limited viewing window inherent in 
computer displays create surprising perceptual misconceptions. 

FIGURE 6. The motion of a piston is 
pictured by a sine graph, which conveys 
certain kinds of information more effec- 
tively than standard algebraic formulas. 



Computer Representation 

Cartesian graphs of numerical and algebraic patterns are only the 
most familiar strategies in an impressive array of visual representations 
for quantitative data. The burgeoning theory of graphs and networks 
includes many new techniques for representing situations with interact- 
ing quantitative and spatial structure. In some cases, network diagrams 
are used to display quantitative information like the costs of shipping 
foods or laying utility lines along various possible paths. In others, nu- 
merical representations such as matrices are used to organize and display 
geometric information like the number of possible paths between nodes 
of a graph. The field of exploratory data analysis includes many other 
new and effective techniques for representing numerical information in 
ways that convey meaning quickly, concisely, and effectively. The use 
of computers to produce those displays is becoming standard practice 
in all areas of applied mathematics. 

One of the principal reasons for using compact symbolic forms to ex- 
press relations among quantitative variables is the marvelous economy 
of capturing the full pattern of many numbers or /7-tuples with a single 
symbolic sentence. However the abstraction required to reduce collec- 
tions of data to symbolic rules also makes the information in those data 
less accessible to many potential users. Fortunately, computer tools also 
make display and reasoning with large data sets easy. 

For example, the difference equation y n +\ = l.0ly n - 445, where 
y Q = 5000, describes the balance of a $5000 loan at 12% interest that 
is being paid back in monthly payments of $445. For most people the 






















































FIGURE 7. Spreadsheet representation of the balance of a $ 5000 
loan at 12% interest that is being paid back in monthly payments 
of $445. 


actual pattern in the dollar value of that loan and the distribution of 
payments to principal and interest is more informatively displayed in a 
simple spreadsheet such as that shown in Figure 7. 

Of course, construction of this spreadsheet requires some ability to 
express relations in the symbolic form that has become standard with 
spreadsheets. In this case the formulas repeat with changing indices as 


Payment Interest Principal Balance 

= 5000.00 

= 445 = 0.01*D2 = A3-B3 = D2-C3 
= A3 = 0.01*D3 = A4-B4 = D3-C4 

Computer-generated numerical representations of algebraic expres- 
sions are proving to be a very useful tool in practical problem solving. 
For instance, to prepare the previous example, we calculated the ap- 
propriate monthly payment by experimental successive approximation, 
not by using the more conventional formula. But these representations 
also serve as a bridge from the concrete world of arithmetic reasoning 
to the more abstract world of algebra and statements that begin "for 

all x " Furthermore, the web of related representations comes full 

circle when computer curve-fitting tools are used to find symbolic rules 
that fit patterns in collections of numerical data. 


The second major aspect of procedural knowledge consists of tech- 
niques commonly referred to as algorithms for using mathematical in- 
formation to solve problems. An algorithm is a "precisely-defined se- 
quence of rules telling how to produce specified output information 
from given input information in a finite number of steps." 23 

Developing student skill in execution of mathematical algorithms has 
always dominated school curricula at both elementary and secondary 
levels. The most prominent algorithms have been procedures for adding, 
subtracting, multiplying, and dividing whole numbers, common frac- 
tions, and decimals, along with the parallel operations on polynomial 
and rational expressions in algebra. But those are only the most ba- 
sic and familiar among a vast library of routine mathematical tools. 
Euclid's algorithm, for example, is only one of several common meth- 
ods for finding the greatest common divisor of two integers; the Sieve 
of Eratosthenes is only one of many algorithms for identifying prime 
numbers; the quadratic formula is one of many algorithms for solv- 
ing quadratic equations; and there are dozens of algorithms for solving 
systems of linear equations and inequalities. 


Design and application of algorithms are obviously at the heart of 
mathematics. The power of mathematics comes from the way that its 
abstract ideas can be applied to solve problems in contexts with no 
surface similarities. The algorithms of irithmetic and algebra that are 
used in business and economics are the same as those used in physics 
and engineering. At the same time the context-independent nature of 
mathematical algorithms makes them easily programmed for computer 
execution. This fact has major implications for school curricula: any 
specific algorithm that is of such fundamental importance and broad 
applicability to merit inclusion in elementary or secondary school will 
certainly have been programmed and made available in standard calcu- 
lator and computer software. Inexpensive calculators can perform most 
numerical, symbolic, and graphic algorithms that are taught in school. 
Thus, current technology seriously undermines any argument that stu- 
dents must develop proficiency in executing any particular algorithm 
because they will need that skill later in life. 

At the same time that learning of specific algorithms has diminished 
in importance for school mathematics, it has become far more impor- 
tant for everyone doing quantitative work to have general understand- 
ing of the algorithmic point of view. 9 ' 23 ' 26 To be an intelligent user of 
computer-based algorithms, it is useful to understand such attributes as 
accuracy, economy, and robustness as well as fundamental mathemati- 
cal concepts like induction and recursion that are too little appreciated 
in traditional curricula. In short, the algorithmic aspect of mathematics 
takes on a very different appearance when calculators and computers 
take over routine systematic procedures. This new condition requires 
fundamental reconsideration of goals for quantitative study in school 

Conceptual and Procedural Knowledge 

Calculators and computers have clearly taken over routine aspects of 
both representation and manipulation of quantitative information the 
two key components of procedural knowledge. The task of translating 
these new conditions into new goals for curricula poses a critical psycho- 
logical question concerning the interplay between conceptual and pro- 
cedural knowledge. Many mathematics educators worry that extensive 
use of calculator and computer tools, with corresponding de-emphasis 
of training in skills, will undermine development of conceptual under- 
standing, proficiency in solving problems, and ability to learn new ad- 
vanced mathematics. 

The interaction of understanding and skill in mathematics has been 
studied and debated intensely for many years but with renewed enthu- 
siasm in the past decade. A recent meta-analysis of over 70 research 


studies 13 concluded that wise use of calculators can enhance student 
conceptual understanding, problem solving, and attitudes toward math- 
ematics without apparent harm to acquisition of traditional skills. More 
limited research in algebra suggests similar conclusions. While there is 
a great deal of work in progress on this issue, the principal reported 
results are from Demana and Leitzel. 8 

However, in almost all of those experiments the calculator or com- 
puter was used to complement instruction in traditional arithmetic and 
algebraic skills. What remains an open and very important problem 
is to determine the consequences of more daring experiments in which 
students are taught to rely more heavily on technological help with arith- 
metic and symbolic manipulation. It seems safe to say that the debate 
over proper consideration of conceptual and procedural knowledge will 
continue for some time. It is certainly the central issue raised by the 
impact of technology in school mathematics. 

Number Sense 

While there is considerable debate concerning the risks and bene- 
fits of shifting attention in school mathematics from traditional skills 
to concepts and problem solving, there is no disagreement about the 
importance of developing student achievement in a variety of infor- 
mal aspects of quantitative reasoning, to develop what might be called 
number sense. Even if machines take over the bulk of computation, it 
remains important for users of those machines to plan correct opera- 
tions and to interpret results intelligently. Planning calculations requires 
sound understanding of the meanings of operations of the character- 
istics of actions that correspond to various arithmetic operations. In- 
terpretation of results requires judgment about the likelihood that the 
machine output is correct or that an error may have been made in data 
entry, choice of operations, or machine performance. (Development of 
number sense is discussed in detail in the February 1989 issue of The 
Arithmetic Teacher, especially in the article by Howden. 16 ) 

There are two fundamental kinds of skill required to test numerical 
results for reasonableness. First is a broad knowledge of quantities in the 
real world: Is the population of the United States closer to 20 million, 
200 million, or 2 billion? Is the speed of an airplane closer to 100, 
1000, or 10,000 kilometers per hour? What are approximate percent 
rates for a sales tax, a car loan, the tip at a restaurant, or success of a 
major league baseball hitter? While this sort of information isn't part 
of formal mathematics, it is an invaluable backdrop for judgment of 
arithmetic applied to real problems. 

The second component of computational number sense is the ability 
to make quick order-of-magnitude approximations. As an electronic 


calculator produces an exact answer, it is important for users to check 
that the displayed results are "in the right ball park." This means, for 
instance, determining by quick rounding and mental arithmetic that 
345 + 257 + 1254 is approximately 1850 or that 85 x 2583 is approx- 
imately 200,000. Skillful mental calculation of this sort is not achieved 
by extensive training in mental execution of traditional written algo- 
rithms, but in flexible application of place value understanding and 
single-digit arithmetic a very different agenda than the goals of tra- 
ditional school arithmetic. Since there has been considerable attention 
given to informal arithmetic and computational estimation over the past 
decade, there now are clear goals, creative curriculum materials, and ef- 
fective teaching suggestions for this important but long-neglected topic. 

Symbol Sense 

There is almost certainly a comparable informal skill required to deal 
effectively with symbolic expressions and algebraic operations to culti- 
vate student symbol sense but ideas and instructional materials in this 
area are not as fully developed. A reasonable set of goals for teaching 
symbol sense would include at least the following basic themes: 

Ability to scan an algebraic expression to make rough estimates 
of the patterns that would emerge in numeric or graphic represen- 
tation. For example, given f(x) = 50*2*, a student with symbol 
sense could sketch the graph of this function and realize that 
function values will be positive and monotonically increasing 
with small values of f(x) for negative x and rapidly increasing 
values for positive x. 

Ability to make informed comparisons of orders of magnitude 
for functions with rules of the form n, n 2 , n 3 , . . . , and k n . This 
skill, a bridge between number and symbol sense, plays an impor- 
tant role in judging the computational complexity of algorithms 
for mathematical and information-processing tasks that are at 
the heart of computer science. 

Ability to scan a table of function values or a graph or to inter- 
pret verbally stated conditions, to identify the likely form of an 
algebraic rule that expresses the appropriate pattern. For exam- 
ple, given the following table, a student with symbol sense could 
predict that the rule for the best-fitting function is likely to be of 
the form f(x) = mx + b with m approximately 15 and b about 

Sales x 10 20 30 40 50 

Costs f(x) 510 675 825 960 1100 1240 


Ability to inspect algebraic operations and predict the form of 
the result or, as in arithmetic estimation, to inspect the result 
and judge the likelihood that it has been performed correctly. 
For instance, a student should realize almost without thinking 
that the product of linear and quadratic polynomials will be a 
cubic polynomial. 

Ability to determine which of several equivalent forms might 
be most appropriate for answering particular questions. For in- 
stance, good symbol sense should allow students to realize that 
the factored form of a polynomial readily yields information 
about its zeroes but makes very difficult calculation of deriva- 
tives or integrals. 

Promising work from current projects shows how numerical and graph- 
ic computer tools can be used effectively to build student intuition about 
algebraic symbolic forms. Nevertheless, the development of more gen- 
eral symbol sense remains an important research task on the path to 
new approaches for developing conceptual and procedural knowledge 
of quantity. 


For a great many students mathematics is a vast, loosely connected 
collection of facts, procedures, and routine word problems. However, it 
is important to remember that the unique power of mathematical con- 
cepts depends on abstract meaning, which lies at the heart of any specific 
embodiment. Learning the fundamentals of any branch of mathemat- 
ics should include recognition of those deep structural principles that 
determine the relations among its concepts and methods. For number 
systems a rather small collection of big and powerful ideas determine 
the structure of each system. When one steps back from specific details, 
it becomes clear that a few central principles govern all algebraic and 
topological properties of numbers. These principles can be used to de- 
rive all specific facts of various number systems and to guide the match 
between formal systems and significant quantitative problems. 

In the historical development of number systems, the progression be- 
gan with the natural numbers. Extensions over many centuries added 
fractions, then negative numbers, and, finally, a rigorous characteriza- 
tion of real numbers. From a perspective near the end of the twentieth 
century it is possible to organize all those structures from the top down: 


The real number system jR is the only complete ordered field. 

The rational number system Q is the smallest subfield of R. 

The integer number system / is the smallest ring in R that in- 
cludes the multiplicative identity. 

The natural number system N is the smallest subset of R that 
includes the multiplicative identity and is closed under addition. 

In the terse form that is characteristic of formal mathematics, these 
four statements contain a great deal of information about structure. 
They imply that each number system is a set with two binary operations 
and a binary order relation; that the operations are commutative and 
associative; that multiplication distributes over addition; that there are 
two identity elements, one for addition and the other for multiplication; 
and that the operations interact with the order relation in familiar ways. 

There are, however, other important properties of the individual num- 
ber systems that are not so apparent from such minimalist characteriza- 
tions. There are significant differences in the algebraic and topological 
properties of the various systems, differences that make each of spe- 
cial interest from both pure and applied perspectives. Analysis of those 
differences, in progression from the simplest to the most subtle, helps 
develop student insight into the nature of numbers and number systems. 
While students should emerge from school mathematics with rich con- 
ceptual and procedural knowledge, it is also important that they have 
some sense of the theoretical principles that provide logical coherence 
to number systems. 

Natural Numbers and Integers 

The fundamental additive, multiplicative, and order structures of the 
natural numbers and integers are based on several simple but powerful 
principles. First is the principle of finite induction: 

If AT is a set of natural numbers that contains 1, and if M con- 
tains the number k 4- 1 whenever it contains the number k, then M 
contains all the natural numbers. 

This property implies that the natural numbers (and their extension to 
all integers) form a discrete set, a sequence of equally spaced elements 
with no number between any integer k and its successor k + 1. They 
provide a set of tags for ordering stages in any process that can be viewed 
as occurring in a sequence of discrete steps. 

The finite induction principle is used to define concepts with integer 
parameters, like x n , and to prove propositions that involve all natural 


numbers. For example, to prove that 1 + 3 + 5 -f . . . + (2n - I ) = n 2 for 
all , one depends on the principle of finite induction: 

1 . Let M be the set of numbers for which the equation is true. Since 
1 = I 2 , we know that 1 e M. 

2. Now suppose that A: eM. Then l + 3 + 5 + ... + (2fc- 1) = k 2 . It 
follows that 1 + 3 + 5 -f . . . + (2k - 1) + (2k + 1) = k 2 + 2k + 1 = 
(fc + I) 2 , so the equation is also true for k + 1. Hence k -f 1 also 
belongs to M. 

3. It follows from the principle of finite induction that M contains 
all the natural numbers, so the formula must be true for all n. 

The method of proof by mathematical induction is used through- 
out mathematics, providing special power in combinatorial propositions 
like the binomial theorem. It has become particularly important as a 
proof technique in computer science, where discrete algorithmic pro- 
cesses are the central objects of study. 

While natural numbers and integers share the discrete order structure 
implied by the principle of induction, there is one critical difference 
between the two systems the existence of additive inverses for integers: 
for every integer a there is an integer -a such that a H a = 0. This 
makes the integers into an additive group, implies that subtraction is 
defined for all ordered pairs of integers, and shows that every equation 
of the form a + x = b has a unique solution in /. 

Although the additive structures of N and / are extremely regular 
and easy to work with, multiplication and division of natural numbers 
and integers hold much more interesting challenges. Since the integers 
contain no multiplicative inverses (except the trivial cases of 1 and 1), 
division is a restricted operation in N and 7, and many equations of the 
form ax = b have no integer solutions. Furthermore, there is no simple 
pattern suggesting which multiplication equations (or related divisions) 
are solvable. The integer 24 is divisible by 2, 3, 4, 6, 8, and 12, but its 
neighbor 23 has no proper factors and 25 has only one proper factor. 
A set of 24 objects can be partitioned into equal subsets in six different 
ways, but a set of 23 cannot be partitioned in any such way. 

Multiplication and division of integers are governed by two principal 
properties. The fundamental theorem of arithmetic guarantees that any 
positive integer can be written as a product of prime factors in exactly 
one way. The division algorithm guarantees that for any positive in- 
tegers a and b there are unique integers q and r such that a = bq + r 
with < r < b. These two principles are of enormous practical and 
theoretical significance in the theory of numbers and, in more general 
form, in algebra. 


The first the prime factorization theorem is one of many similar 
results in mathematics showing how complex expressions can be studied 
effectively when they are written as a combination of irreducible factors. 
These applications range from the mundane task of finding least com- 
mon multiples or greatest common factors to the parallel fundamental 
theorem of polynomial algebra which assures that any polynomial with 
complex coefficients can be written as a product of linear factors (from 
which the zeroes can be easily obtained). 

The division algorithm is, of course, basic to the familiar procedure 
for long division of natural numbers and decimals as well as to the 
parallel factor theorem of polynomial algebra. It provides the essential 
concept for developing the arithmetic of congruences: For any integers 
a and b,a = b (mod m) if and only if a = mk -f b for some k. The finite 
cyclic groups and fields that arise from this theory have proven useful in 
dramatic ways as models for discrete phenomena, including increasingly 
important applications in computer science, in cryptography, and in 
transmission and storage of business and governmental information. 

Rational Numbers 

The smallest number system that includes elements representing each 
possible division of integers a/b (for nonzero b) is, of course, the 
rational number system Q. Mathematicians call Q a field, a term used 
to describe other structures with similar number-like properties. In Q 
every nonzero element has a multiplicative inverse, and every linear 
equation of the form rx 4- s = t has a unique solution for rational r, s, 
and t (for nonzero r). However, this algebraic power is gained at the 
expense of simplicity. 

The standard ordering of rational numbers makes them a dense set 
between any two rational numbers there is a third rational number. In 
particular, there are positive rational numbers as small as one might 
wish. On the other hand, for any rational numbers a and /?, there is an 
integer n such that na > b\ this property makes the rational numbers 
into an Archimedean ordered field. While the operations and ordering of 
rational numbers are significantly more complex than integers, the den- 
sity and Archimedean properties of Q combine to lay the groundwork 
for precision in measurement, guaranteeing that a unit of any desired 
refinement can be used to cover a length of any finite extent. 

Real Numbers 

The natural numbers, integers, and rational numbers provide for- 
mal systems to model the structures of many practical quantitative 
reasoning tasks. But unresolved questions raised as long as 2000 years 




FIGURE 8. The position on a rational number line 
corresponding to the length of the hypotenuse of a right 
triangle with legs of length 1 has a hole, since there is 

{2 no rational number equal to \/2. 

ago make it quite clear that the rational numbers are not the last word 
in number systems. The proof that there is no rational number whose 
square is 2 (or 3, or 5, or any other integer that is not a perfect square) 
reveals an algebraic incompleteness in the rational number system (see 
Figure 8). When numbers are used as measures of geometric figures, 
the Pythagorean theorem shows that there are line segments with no 
rational measures. There are "holes" (although not very big holes) in a 
number line that has only rational coordinate points. 

The rational numbers can be extended in a variety of ways to include 
elements that fill some of these holes and that fulfill specific algebraic or 
geometric needs. The extension Q(V2) = {a + bV2:a,b e Q}, for in- 
stance, is an ordered field, under a suitable definition of addition, multi- 
plication, and inverses. However, the only complete ordered field one 
that fills all the holes is the real number system R. It is an ordered 
field in which every nonempty subset that is bounded from above has 
a least upper bound in R. A key theorem of number systems, one that 
establishes a distinctive role for R, is that any such complete ordered 
field must be isomorphic to .R. 

Since the real numbers seem only to fill "infinitesimal" holes on the 
rational number line, several other differences between the two number 
fields are genuinely surprising. First, while every rational number is 
the solution of a simple equation ax = b where a and b are integers, 
there are transcendental real numbers (like e and ri) that are not solu- 
tions of any such polynomial equation. Furthermore, while the rational 
numbers can be placed in one-to-one correspondence with the natural 
numbers and are thus countably infinite (a surprising result that was not 
comfortably understood until early in this century), this is not true for 
the real numbers. In fact, the transcendental numbers alone are more 
numerous than the algebraic numbers those that arise as solutions to 
rational algebraic equations. While this last result was proven at least 
100 years ago through very clever reasoning with transfinite cardinal 
numbers, there are still subtle outstanding questions about the charac- 
ter of specific real numbers. 

The real numbers provide a significant step in the development of 
quantitative concepts and methods in another fundamental sense. While 
the natural numbers, integers, and rational numbers are each infinite sets 


of numbers, their primary use is to count, order, and compare finite sets 
of discrete objects. The real numbers provide the essential mathemati- 
cal tool to describe and reason about infinite and infinitesimal processes. 
They alone support rigorous development of the concepts of limit and 
continuity; they provide the bridge to analysis of motion and change. 

Complex Numbers 

The extension from rational to real numbers enables solution of many 
simple and significant algebraic equations. But it leaves an equally sig- 
nificant collection of algebraic equations still unsolvable. Simple poly- 
nomial equations like jc 2 + 1 = or x 2 +x+ 1 = have no real roots. The 
number system required to give meaningful solutions to these equations, 
and to all polynomial equations in general, is the complex numbers C. 
The complex numbers constitute the smallest possible field extension of 
the real numbers that contains an element i with square equal to -1, 
the required root of x 2 + 1 = 0. Remarkably, the extension to deal with 
this single equation provides solutions to all other polynomial equations 
and opens a rich structure of mathematical properties and applications. 

Every complex number can be expressed in the form a + bi, where a 
and b are real numbers. Thus the complex numbers are determined by 
ordered pairs of real numbers. While the real numbers can be ordered 
in one-to-one correspondence with the points of a line, the complex 
numbers correspond to points of a two-dimensional plane and are not 
linearly ordered. This loss of simple order might seem to promise a 
much more complicated life in C than in the real numbers or their 
subsets. However, it brings along benefits as well. The correspondence 
between complex numbers and points in the plane opens a powerful 
connection betweeen the arithmetic and algebra of C and the geometry 
of shapes and transformations in the plane (see Figure 9). 

The complex numbers include some numbers originally described as 
"imaginary" by mathematicians who could not admit the possibility 
of a negative square. Nevertheless, they have proven useful as mod- 
els of many very real physical phenomena, from the flow of alternating 

FIGURE 9. Points in the plane corre- 
spond to complex numbers, with addition 
of vectors in the plane reflecting addition 
of complex numbers. Multiplication is 
more complicated the magnitudes of the 
vectors multiply as expected, but the angles 


electrical current to the flow of air over an airplane wing. They also settle 
a fundamental algebraic question of pure mathematics: every polyno- 
mial of degree n has exactly n linear factors. Thus every polynomial 
equation has at least one and at most n distinct complex roots. 

New Number Systems 

Our sketch of fundamental principles of number systems covers very 
familiar ground. When mathematicians of the late nineteenth century 
showed that the real number system is the unique complete ordered field, 
following earlier proofs by Gauss and others that the complex number 
system is algebraically closed, it seemed that the story of number systems 
was complete. While that is, in some sense, an accurate statement, the 
development of new number systems is by no means finished. 

For example, since their invention in the mid-nineteenth century, the 
algebra of matrices has become an invaluable tool for reasoning about 
complex numerical data. A matrix is a kind of super-number; within 
certain families of matrices, the operations of addition and multiplica- 
tion have algebraic properties very similar to those of the real numbers. 
The most prominent exception is the fact that matrix multiplication is 
noncommutative a fact that has many important consequences in the 
theory of linear algebra. Matrices are particularly useful for describing 
complex sets of quantitative data such as those that computers routinely 

The application of computing to quantitative reasoning has stim- 
ulated development of mathematical systems in another direction of 
both practical and theoretical interest. Despite their seemingly endless 
memory and instantaneous speed, computers work not with the famil- 
iar number systems such as /, ), or j?, but in finite approximations of 
those systems whose faithfulness is limited by the ability of computer 
languages to represent numbers with only a finite number of positional 
places. These "truncated" models of number systems do not obey the 
conventional structural properties of numbers (such as associativity of 
addition). Thus it seems important that students extend their study to 
include the structural properties of those finite systems that underlie so 
much of their actual quantitative work. 

The discovery of number-like mathematical systems like matrices that 
fail to obey structural properties that our naive intuition tells us are 
true was a dramatic step in the development of modern mathematics. 
Contemporary algebra originated in an attempt to provide a theory to 
explain the structural properties of various number systems. In the 
last 150 years algebra has generated a rich array of abstract theories 
that spring from study of structure inherent in various operations and 


relations on sets. Mathematicians have shown that generalization of 
number systems can provide a stimulating intellectual playground. But 
they have also shown that this abstract mathematical realm frequently 
has impressive practical applications. Although groups, rings, fields, 
lattices, boolean algebras, monoids, and Turing machines were created 
primarily as abstract possibilities, they are now used routinely as tools 
for research on fundamental problems of computing and information 

During the middle of this century, mathematics was strongly influ- 
enced by interest in exploring generalizations of number systems. In 
1973 Garrett Birkhoff 3 wrote that, "by 1960 most younger mathemati- 
cians had come to believe that all mathematics should be developed 
axiomatically from the notions of set and function." Furthermore, he 
and MacLane, "wrote another 'Algebra' which went further in the di- 
rection of abstraction, by organizing much of pure algebra around the 
central concepts of morphism, category, and 'universality'." Innovative 
school mathematics programs of the 1960s explored the possibilities of 
organizing curricula around similar abstract structural concepts. 

Fashions change, in mathematics as well as in design of human ar- 
tifacts. Today the abstract axiomatic point of view seems much less 
promising as a guide to either mathematics research or education. None- 
theless, there are central principles that lie at the heart of number sys- 
tems and algebra. They provide coherent organization for what can be 
an impenetrable maze of specific facts and techniques, and this organi- 
zation is as useful for students as for practicing mathematicians. Thus 
it seems wise for curriculum planners to identify and build from such 
principles as they plan school curricula. 


School mathematics must develop in students an understanding of 
basic principles, proficiency in techniques, and facility in reasoning. But 
the ultimate test of school mathematics is whether it enables students 
to apply their knowledge to solve important quantitative problems. The 
ability to solve problems is not only the most important goal of school 
mathematics but also the most difficult educational task. 

The term ''word problem" strikes terror in the hearts of mathematics 
students of aU ages. The key first step in effective work on problems 
is to identify in problem situations concepts that are structurally sim- 
ilar to those of number systems. Traditional approaches to this task 
can be sorted into two broad classes. The pragmatic approach helps 
students cope with a variety of classical (and nearly routine) problem 
types. The aim is to provide students with strategic guidelines for 


each problem type a special chart for organizing information about 
time/rate/distance problems, a dictionary for translating key quantita- 
tive words into symbolic expressions, and so on. The more ambitious 
approach attempts to train students to use generic high-level strategies 
(or heuristics) that apply to problems in many different areas. 

It seems fair to say that neither approach is demonstrably effective 
in providing students with confident and transferable modeling and 
problem-solving skills. The (unfortunately) popular "key words" ap- 
proach fails because the flexible, versatile, and often ambiguous 
structure of ordinary language cannot be translated into mathematical 
statements by any dependable algorithm. At the other extreme, while 
students can learn generic high-level heuristics suggested by Polya and 
others, it has proven very difficult to develop their facility in the kind 
of metacognitive monitoring of thought that is required to deploy those 
heuristics effectively in specific situations. Recent work to develop a 
metacognitive perspective on problem-solving strategies shows promis- 
ing but not yet definitive results. 30 


While the search continues for effective new strategies to teach prob- 
lem solving, there is an equally significant change emerging in think- 
ing about the nature of quantitative problems themselves. In many 
contemporary applications of mathematics one thinks less about solving 
specific well-defined problems and concentrates instead on constructing 
and analyzing mathematical models of the problem setting. The clas- 
sical quantitative problems of school mathematics usually include nu- 
merical information and a single question that can be answered by a 
numerical calculation or by solving an equation. Outside school, prob- 
lem situations generally have missing or extraneous information as well 
as many ill-defined questions. 

In a mathematical modeling approach, the first step is to identify rel- 
evant variables. The next is to describe, in suitable formal language, 
relations that represent cause-and-effect connections among those vari- 
ables. Specific questions can then be posed in terms of input or output 
values or global properties of the modeling relations. Finally, computer 
tools can be used to answer those questions by numerical, graphic, or 
symbol methods. 


The most common sources of numerical variables are measurements. 
Thus the theory and technique of measurement play important roles in 


quantitative literacy. Like the arithmetic of number systems, measure- 
ment feels like a familiar and well-known facet of school mathematics 
hardly in need of new thinking. However, this critical interface between 
mathematics and its applications is not a remarkably successful topic in 
the curriculum. 

The prototypical measurement tasks in school mathematics are find- 
ing length, area, and volume of geometric figures. It seems fair to say 
that for most students learning about measurement includes brief ex- 
posure to a few standard units for length and then practice in use of 
formulas for perimeter, area, and volume based on those length units. 
Area is [length x width] or [(1/2) x base x height] or [n x r 2 ], volume 
is [length x width x height] or [n x r 2 x h] 9 and so on. Most exercises 
become arithmetic practice in the formula of the lesson at hand. 

Students exposed to this formal approach to measurement generally 
form limited and very rigid conceptions of length, area, and volume. 
Confusion of area and perimeter is a depressingly common error on 
student assessments. The common "rule" followed by unthinking stu- 
dents, regardless of any wording in the problem statement, is that if 
there are two numbers attached to sides of a pictured rectangle one 
multiplies them; if there are numbers on each side of a rectangle, one 
adds them. 

The emphasis on formulas also leaves students ill prepared to deal 
intelligently with the approximate nature of real measurements, the cu- 
mulative effects of errors in combinations of measurements, and the 
generalization to irregular shapes that occur in so many practical appli- 
cations or to the curves and surfaces that are fundamental in calculus. 
Furthermore, few students realize or take advantage of structural simi- 
larities that underlie most applications of measurement. 

At the heart of any measurement process is a mapping that assigns 
numbers to objects. The mapping assigns measure 1 to some designated 
unit. Other objects are then covered by copies of the unit. The choice 
of unit element is arbitrary, but once the choice is made, it provides the 
standard by which all others are measured. Thus every measurement 
consists of a unit and a number the number of whole and partial copies 
of the unit needed to exactly cover the measured object. The mathe- 
matics student who understands this principle as a general property of 
many important measurements has acquired productive insight into 
the connection between real situations and quantitative models. 

The unit and covering properties of measurement explain quite clearly 
just what is being indicated by any particular measurement; moreover, 
the attachment of units to measurements can be exploited to guide for- 
mal reasoning about scientific principles. In many sciences quantitative 
reasoning is guided by a well-defined algebra of quantities commonly 


called "dimensional analysis." In this method each arithmetic opera- 
tion is performed not only on numbers but on the units as well. If the 
end result is a number whose units are appropriate for the problem, the 
dimensional analysis lends support to the appropriateness of the oper- 
ations that have been performed. While this attention to units as well 
as numbers in measurement is not as common in mathematics as in 
science, it has strong supporters among those who have been concerned 
with helping students make the connection between formal mathematics 
and its applications. 18 ' 21 ' 31 

The theory and practice of measuring quantitative concepts in the 
physical world have a long history in mathematics and its teaching. 
However, just as many classical mathematical methods have been gen- 
eralized and applied in new domains, measurement has been extended 
to important uses throughout the social sciences. While the basic idea 
is the same assigning numbers to objects or events these new mea- 
sures often obey structural properties that are very different from the 
measures of length, area, and volume. 

Political scientists and sociologists have designed a variety of mea- 
sures of influence or power in social situations. Economists have de- 
vised measures of costs and benefits to quantify options in decision 
making. Psychologists and educators use a vast array of measures to 
describe aptitudes and achievements of individuals. Statisticians mea- 
sure probable cause-and-effect relations among many different kinds of 
stochastic variables. In each case, numbers, operations, and relations 
are used to model significant structural properties of situations. Some- 
times classical principles and concepts are directly applicable. But it is 
increasingly common that effective quantitative reasoning in the social 
and human sciences requires understanding of aspects of number that 
permit flexible construction of new responses to new situations. 


Without question the most important goal of school mathematics is 
to develop students' ability to reason intelligently with quantitative in- 
formation. The mathematical concepts, techniques, and principles that 
model quantitative aspects of experience are provided by structures of 
number systems, algebra, and measurement that have long been the 
heart of school curricula. However, the emergence of electronic calcu- 
lators and computers as powerful tools for representing and manipu- 
lating quantitative information has challenged traditional priorities for 
instruction in those subjects. It no longer makes sense to devote large 
portions of the school curriculum to training students in arithmetic or 
algebraic algorithms that can be performed quickly and accurately by 


low-cost and convenient calculators. The availability of powerful aids 
to computation has also led to a dramatic increase in the range of sit- 
uations to which quantitative reasoning is being applied. Thus school 
mathematics must prepare students to use their knowledge of number, 
algebra, and measurement in flexible and creative ways not only in 
routine, predictable calculations. 

To prepare students for the challenge of quantitative reasoning in the 
modern world, school mathematics must develop students' abilities to 

Understand fundamental properties of number systems and the 
match between those mathematical systems and real-life situa- 
tions in which they are embodied. 

Describe and interpret quantitative structures using symbolic, 
verbal, and graphic representations. 

Perform both exact and approximate calculations involving arith- 
metic and algebraic ideas by various appropriate methods men- 
tal operations, paper-and-pencil techniques, calculators, or com- 

Apply numerical and algebraic expertise to solve both routine 
and original quantitative problems. 

The school experience likely to develop these general skills and under- 
standings must be rich in opportunities to explore interesting and im- 
portant quantitative situations as well as in the structures that illuminate 
mathematical ideas embodied in specific settings. 


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"Uncertainty" is intended to suggest two related topics: data and 
chance. Neither is a topic within mathematics; they are both, however, 
phenomena that are the subject of mathematical study. Roughly speak- 
ing, statistics and probability are the mathematical fields that deal with 
data and chance, respectively. 

Recent recommendations concerning school curricula are unanimous 
in suggesting that statistics and probability should occupy a much more 
prominent place than has been the case in the past. 12 ' 14 However, be- 
cause of the emphasis that these recommendations place on data anal- 
ysis, it is easy to view statistics in particular as a collection of specific 
skills (or even as a bag of tricks). The task of this essay is not to urge 
attention to data and chance in the school curriculum they are already 
attracting attention but to develop this strand of mathematical ideas 
in a way that makes clear the overall themes and strategies within which 
individual topics find their natural place. 

Any discussion that is intended to influence teaching should reflect 
the experience of teachers and students. Suggestions for curriculum re- 
form detached from that experience offer Utopian hopes that are disap- 
pointed in practice. Statistics in the schools is not Utopian; new material 
presently being tested is practically useful and aids rather than displaces 
development of number concepts and skills. Nonetheless, it is easy in 
our enthusiasm to overlook practical problems and to urge the teaching 


of subject matter that is unrealistic in quantity or level. It is important 
to call attention to the difficulties and potential false steps, as well as to 
the advantages, in using data and chance in the teaching of mathemat- 
ics. In writing this essay I have tried to err in the practical rather than 
the Utopian direction. 


Interest in teaching statistics is certainly due in part to recognition 
of the place that working with data plays in everyday life and in many 
occupations. It is increasingly common to teach mathematical topics 
that are of direct use, rather than to select topics simply because they 
lead to later topics in mathematics. Statistics is such a topic. 

News reports present national economic and social statistics, opinion 
polls, medical data from both epidemiological studies and clinical trials, 
and business and financial data. Many citizens must deal with data in 
more detail on the job. Fanners and agribusiness use crop forecasts and 
the results of agricultural field trials. Engineers are concerned with data 
on product performance, quality, and reliability. Manufacturing work- 
ers are increasingly asked to record and act on data for process control. 
The health sciences struggle with data on cost and effectiveness as well 
as with data from medical research. Business runs on data of every va- 
riety: costs, profits, sales projections, market research, and much more. 
There are compelling practical reasons to learn statistics. 

As these examples suggest, data are not merely numbers, but numbers 
with a context. The number 10.3 in the absence of a context carries 
no information; that the birth weight of a baby is 10.3 pounds enables 
us to comment on the healthy size of the child. That is, data engage 
our knowledge of their context so that we can understand and interpret, 
rather than simply carry out arithmetical operations. 

There are, therefore, strong pedagogical as well as practical reasons 
to teach statistics in the schools. Statistics combines computational ac- 
tivity in a meaningful setting with the exercise of judgment in choosing 
methods and interpreting results. Statistics in the early grades is taught 
not primarily for its own sake, but because it is an effective way to de- 
velop quantitative understanding and to apply arithmetic and graphing 
to problem solving. 

Teachers who understand that data are numbers in a context will al- 
ways provide an appropriate context when posing problems for students. 
Calculating the mean of five numbers is an exercise in arithmetic, not 
statistics. Calculating the mean price of a popular music tape at five 
retail outlets is statistics, particularly when combined with a look at the 
spread in the prices and a comparison with the price of other types of 


It is essential that the practical and pedagogical advantages of work- 
ing with data not succumb to an exclusive emphasis on teaching oper- 
ations. Teachers and developers of curriculum material must exercise 
imagination in providing data that are meaningful to students. In the 
upper grades, data from other academic subjects (such as science) can 
be used, although students rarely connect such data with their every- 
day life. In the lower grades, data produced by the students themselves 
are best. Students can produce data in many ways, such as questioning 
the class ("How many children live in your house?") or by asking each 
student to measure, count, or estimate some quantity. 

The additional effort required to provide data rather than simply num- 
bers should be taken into account when planning instruction. Good 
data are not just an attractive feature for motivating students; they are 
essential to the nature of statistics. Yet it is important that the effort 
required to produce data not overshadow the mathematical ideas taught 
and learned. 

In particular, attempts to produce good data on important issues out- 
side school are always much more difficult than is at first apparent. 
Unpleasant experiences with time-consuming and confusing attempts 
to produce data may well discourage teachers from teaching statistics. 
The difficulties associated with data production activities form the first 
of several potential barriers to effective reform. Curriculum materials 
must provide both interesting data and practical, tested suggestions for 
production of data by students. Over time, teachers can collect and 
share data sets that pertain to their community and school. Computers 
are an ideal means of storing and sharing data. 


Some phenomena have predictable outcomes: drop a coin from a 
known height and the time it takes to fall can be predicted from basic 
physics. Except for a rather small measurement error, the outcome 
is certain. If we toss the coin, on the other hand, we cannot predict 
whether it will show heads or tails. The outcome is uncertain. Yet 
coin tossing is not haphazard. If we make a large number of tosses, 
the proportion of heads will be very close to one-half. This long-term 
regularity is not just a theoretical construct but an observed fact: 

* The French naturalist Buffon (1707-1788) tossed a coin 4040 
times. Result: 2048 heads, a proportion of 2048/4040 = 0.5069 
of heads. 

Around 1900 the English statistician Karl Pearson heroically 
tossed a coin 24,000 times. Result: 12,012 heads, a proportion 
nf n 


9 The English mathematician John Kerrich, while imprisoned by 
the Germans during World War II, tossed a coin 10,000 times. 
Result: 5067 heads, a proportion of 0.5067. 

Phenomena having uncertain individual outcomes but a regular pat- 
tern of outcomes in many repetitions are called random. "Random" is 
not a synonym for "haphazard" but a description of a kind of order 
diiferent from the deterministic one that is popularly associated with 
science and mathematics. Probability is the branch of mathematics 
that describes randomness. 

The experience of children in and out of school provides less contact 
with randomness than with data. For example, students do not meet 
areas of science in which random behavior appears (such as genetics 
and quantum theory) until secondary school and then only if they elect 
solid science courses. Uncertainty is of course a pervasive aspect of all 
human experience; it is the order in uncertainty that is hard to observe in 
casual settings. Even state lotteries, although familiar to many students, 
give little experience with the orderly aspect of randomness because of 
their emphasis on extremely unlikely large prizes. These well-publicized 
games of chance use actual physical randomization but appear to make 
people rich haphazardly. 

Psychologists have shown that our intuition of chance profoundly 
contradicts the laws of probability that describe actual random behav- 
ior. This incorrect understanding is very difficult to correct by formal 
instruction. Attempts to teach probability and statistical inference with- 
out adequate intuitive preparation are a second major pitfall in intro- 
ducing data and chance into school curricula. 

Even at the college level many students fail to understand probability 
and inference because of misconceptions that are not removed by study 
of formal rules. The conflict between probability theory and students' 
view of the world is due at least in part to students' limited contact with 
randomness. We must therefore prepare the way for the study of chance 
by providing experience with random behavior early in the mathematics 
curriculum. Fortunately, the study of data provides a natural setting for 
such experience. The priority of data analysis over formal probability 
and inference is an important principle for instruction in uncertainty. 

Artificial chance devices (coins, dice, spinners) can be used to pro- 
duce data in the classroom with the intent of applying data analysis 
skills to discover the orderly nature of these devices. Uncertainty also 
appears in data from sources other than chance devices. Repeated mea- 
surements of the same quantity (made by several students, for example) 
yield varying results. Natural variation appears in the heights, reading 
scores, or incomes of a group of people. It is perhaps surprising that 


the patterns of variation in careful measurements or in data on many 
individuals can be described by the same mathematics that describes 
the outcomes of chance devices. 

Experience with variation in data is a first step toward recognizing the 
connection between statistics and probability. At a later stage the role 
of deliberate randomization in statistical designs for producing data 
strengthens this connection. Finally, formal statistical inference uses 
the language and facts of probability to express the confidence we can 
have in conclusions drawn from data. 

Although the usefulness in everyday life of an understanding of ran- 
domness is less obvious than the necessity of dealing with data, prac- 
tical arguments for teaching about chance are not absent. One goal of 
instruction about probability is to help students understand that chance 
variation rather than deterministic causation explains many aspects of 
the world. 

Suppose that a basketball player over a long season has made 
70% of her free throws. At the end of a tournament game she 
attempts five free throws and makes only two. "Nervousness," 
say the fans. But this causal explanation need not be correct. 
A player having a probability of 0.7 of making each shot has a 
probability of about 0. 16 of missing three or more of five shots. 
Such a performance can easily be simply chance variation. 

Some understanding of probability enables us to consider the role of 
chance rather than seek a specific cause, oftentimes spurious, for every 

Calculators and Computers 

While the advent of fast, easily accessible computing has had an im- 
pact on mathematics as a whole, it has revolutionized the practice of 
statistics. An obvious effect of the revolution is that more complex anal- 
yses on larger sets of data are now easy. But the computing revolution 
has also brought about changes in the nature of statistical practice. In 
the past statisticians conducted straightforward but computationally te- 
dious analyses based on a specific mathematical model in order to draw 
conclusions from data. Instruction in statistics showed a corresponding 
emphasis on learning to carry out lengthy calculations. 

Now the paradigm statistical analysis is a dialogue between model 
and data. The data are allowed to criticize or even falsify the origi- 
nal model. Diagnostic methods to aid this process are a major field 
of research in statistics. All are computationally intensive, and the 
most widely adopted make heavy use of graphic display. In addition, 


freedom from the limits once imposed by hand calculation has led to 
new methods for inference from even quite small data sets. 3 This chang- 
ing nature of statistics is readily reflected in instructional styles, espe- 
cially in increased emphasis on graphical methods and informal data 

The influence of computers has led to some soul searching among 
mathematicians, some of whom question the nature of a proof based on 
a computer search of possible cases too numerous for human scrutiny. 
At a more elementary level, both teachers and parents ask whether early 
use of calculators will impede understanding of numbers and arithmetic 
operations. Statisticians, on the other hand, have welcomed calculators 
and computers as a liberating force. Calculating sums of squares by 
hand does not increase understanding; it merely numbs the mind. In 
these circumstances it is natural for a statistician to urge the use of 
calculators and computers in instruction about data at all levels. 

College teaching of statistics already makes universal use of calcula- 
tors and wide use of statistical software on computers. (There is, of 
course, a continuum rather than a disjunction between calculators and 
computers as technology continues its advance.) Here is a typical exer- 
cise from basic statistics, reconsidered in the light of easy computing. 

Figure 1 presents a scatterplot of data on the age at which 
each of a group of children spoke their first word and their later 
scores on a test of mental ability. Does age at first word help us 
predict the later test score? 

Once upon a time a student would be asked to plot the data and then 
calculate the least squares regression line (the solid line in Figure 1) 
together with the correlation coefficient r = -0.640. Perhaps the plot 
would be omitted to save time. Most students would require at least 1 5 
minutes for this exercise with a basic calculator. Only a sadist would 
ask much more of them. 

But it is apparent that the data include two outliers, labeled as cases 1 8 
and 19 in the plot. How do these cases influence the regression analysis? 
An interactive software package of the kind that is widely available on 
all varieties of computers provides immediate answers, which can be 
visually displayed if the computer has graphics capabilities. Case 19, 
although far from the regression line, does not have a large influence 
on the position of the line or the value of the correlation r. Case 18, 
on the other hand, is highly influential. Removing this point moves the 
regression line to the dashed line in the figure and reduces the correlation 
to r = -0.335, about half its original value. Thus the evidence that age 
at first word predicts later ability scores is much weaker if case 1 8 is 




"J 120 









Case 19 

O= 2 cases 








FIGURE 1. Data on the age at which each of 21 children first spoke (horizontal scale) 
and their Gesell Adaptive Score, the result of an aptitude test taken at a later age. Case 
1 8 is particularly influential in the sense that deleting this point substantially moves the 
regression line and changes the value of numerical measures such as the correlation. 

dropped. (These data are discussed in detail in Examples 3.10 and 3.14 
of Moore; 13 most of the figures in this essay are drawn from that text.) 

Automating the calculations preserves our energy for a discussion of 
the data. It is natural for the discussion to take the form of group prob- 
lem solving: "Is anything unusual? Outlying points. How important 
are they? Let's try doing the analysis again without them." We are 
then encouraged to seek additional information about the context of 
the data to ask, for example, if the child of case 1 8 is so slow to begin 
talking as to be out of place in a study of normal child development. 
The example also leads us to ask what makes an observation influential, 
a question that leads to new and important subject matter in statistics. 

Automated calculation allows students to concentrate on other aspects 
of problem solving: planning an appropriate analysis, interpreting the 
results in their context, and asking new mathematical questions sug- 
gested by an exercise. But it is also true that automated calculation can 
hide the nature of the work that is carried out and impede judgment 
about whether the work was appropriate to this specific problem. Too 
often, students believe that computers simply inform us about the truth, 
as in the Star Wars movies. 

In a classroom exercise on sampling, 18 students were asked to record 
the colors of a large sample of M&M candies and to compare the 


results with computer-produced samples from a uniform distribution of 
the same colors. The distribution of colors in the candies was far from 
uniform. The purpose of the exercise was to demonstrate from the com- 
parison that the candy colors were not, in fact, uniformly distributed. 
Yet "... some students simply believed that the computer model was 
correct because it was on the computer, even though they had entered 
the population model themselves." 

Overoptimism about the effectiveness of computers is a major poten- 
tial pitfall in teaching statistics, as is insufficient planning to integrate 
calculators and computers into the curriculum. Graduated use of calcu- 
lators and computers is essential if students are to gain their advantages 
without coming to believe in a "magic box." 

Basic arithmetic skills are needed for mental arithmetic and estima- 
tion, which are important in checking automated calculations. Four- 
function calculators preserve control over the order of operations, which 
must be requested one by one, while automating only the algorithms. 
A child must understand, for example, the distinction between divisor 
and dividend in order to use a calculator for long division. A child must 
know that one finds a mean by adding the observations and dividing by 
their number in order to compute jc with a basic calculator. Children 
can therefore begin to use calculators in their study of data as soon as 
the operations are understood. Later, a calculator that will compute the 
sample mean and standard deviation directly from keyed-in data can 
be used to bypass routine algorithms already mastered. 

At a more advanced level, some histograms should be made by hand 
before turning to attractive software that chooses groups and creates 
histograms directly from the raw data. Perhaps most importantly, ex- 
perience with physical chance devices and physical simulations such as 
drawing colored beads from a box should precede computer simulations. 
"Microworlds" need have no connection with reality, yet students tend 
to believe that the computer presents reality. A carefully graduated 
transition from physical to digital is very important. The practice of 
graduated use is easiest when calculators and computers are part of the 
normal classroom environment to be used as needed, not reserved for 
special projects or upper grades. 

From Data to Inference 

There are several organizing principles that help us see the mathemat- 
ical study of data and chance as a coherent whole. One such principle 
is the progression of ideas from data analysis to data production to prob- 
ability to inference. The discussion in this essay is organized in these 
same stages: 


Data analysis, which involves organizing, describing, and sum- 
marizing data. 

* Producing data, usually to answer specific questions about some 
larger population. 

* Probability, the mathematical description of randomness. 

Inference, the drawing of conclusions from data. 

This progression of topics represents both the logical development of 
the field and the level of difficulty of the concepts. It therefore gives 
the general order in which statistical topics should appear in the school 
curriculum. Of course, the latter three headings will appear informally 
from the beginning in the context of data analysis. Experience in pro- 
ducing data in particular, experience with chance outcomes can be- 
gin in the earliest grades. Similarly, informal conclusions based on data 
should be encouraged from an early stage. 

The main drawback to this outline is that it does not emphasize that 
probability is important in its own right, not merely as a part of statis- 
tics. Both the concept of probability and basic mathematical facts about 
probability can be introduced in elementary school as soon as fractions 
are understood. There is, however, a natural place for probability in the 
progression of statistical ideas. Statistical designs for producing data are 
characterized by the deliberate use of chance in random sampling and 
randomized comparative experiments. Here is an opportunity to pro- 
vide more experience with randomness and to advance to a study of 
random variation in numerical summaries (such as the mean of several 
observations). Both physical random selection and simulation can be 

On the other hand, formal statistical inference requires some under- 
standing of probability. Therefore it makes sense that the section on 
probability be between those on producing data and inference. Because 
of the great conceptual difficulties that students encounter in probability 
and in probability-based inference, formal mathematical treatment of 
these subjects should probably be an elective rather than a core course 
in secondary school. 


Data analysis is descriptive statistics reborn, with new methods, 
greater emphasis on graphics, and a consistent philosophy due to John 
Tukey. (Volumes 3 and 4 of Tukey's Collected Works contain his writ- 
ings in this area. 8 A reviewer recommends paper 12 in Volume 4 as 
a good starting point.) The essence of data analysis is to "let the 


data speak" by looking for patterns in data without at first consider- 
ing whether the data are representative of some larger universe. 

Inspection of data often uncovers unexpected features. If the data 
were produced to answer a specific question this is the setting in which 
such traditional methods as confidence intervals and significance tests 
are best justified the unusual features may lead us to reconsider the 
analysis we had planned. Careful data analysis therefore precedes for- 
mal inference in good statistical practice. 

In other cases we do not have specific questions in mind and want 
to allow the data to suggest conclusions that we can seek to confirm 
by further study. We then speak of "exploratory data analysis," on the 
analogy of an explorer entering unknown lands. 

The best-known contributions of data analysis are new methods for 
displaying data, such as stemplots and boxplots (or stem-and-leaf plots 
and box-and-whisker plots if you prefer longer terms). From these ex- 
amples it is easy to see data analysis as a collection of clever tools and 
miss the organizing principles. Both analyses of complex data sets and 
the order of instruction about data can usefully be guided by three sim- 
ple principles: 

1. Move from simple to complex, from examining a single variable 
to relations between two variables and connections among many 

2. When examining data, look first for an overall pattern and then 
for marked deviations from that pattern. 

3. Move from graphic display to numerical measures of specific as- 
pects of the data to compact mathematical models for the overall 

Displaying Data 

The first and third principles suggest that learning about data starts 
with displaying the distribution of a single variable. Most such data are 
either counts that is how qualitative variables such as color become 
numerical or measurements with units. Specific methods for data dis- 
play can advance in parallel with the development of early quantitative 
concepts. "How many of each color in a bag of M&Ms?" can be deter- 
mined by counting and displayed with stacks of colored blocks. 

Later a stemplot of two-digit numbers can reinforce the distinction 
between the 10's and the 1's place in whole numbers. A stemplot of two- 
digit data lists each 10's digit as a "stem" and records the observations 
by placing their Ts digits as "leaves" on the appropriate stem. Here, 
for example, is a stemplot of the number of home runs Babe Ruth hit 
in each of his years with the Yankees. 








Still later we come to histograms. To construct histograms of data with 
more than a few values requires an understanding of "betweenness" and 
the ability to group numbers, as well as skill in making and using scales 
on graphs. 

Choice among the available variations on stemplots and histograms 
requires more judgment as the numbers making up the data become less 
simple. Stemplots of numbers with several digits often require round- 
ing or truncation, for example. Grouping numbers with several decimal 
places into classes for a histogram requires a clear understanding of 
order for decimal numbers. Careful planning is important to avoid in- 
advertently presenting students with tasks that go beyond their number 
skills. But it is also clear that data analysis in the elementary grades can 
reinforce important concepts and skills from the existing mathematics 
curriculum by applying them in interesting settings. 

When we have constructed a display, we must interpret it and com- 
municate our understanding to others. Children are not naturally able 
to "read" data any more than they are born able to read words. They 
must be taught both the strategy of looking at data and specific features 
to be aware of. The strategy is expressed in the second principle: look 
for pattern, then for deviations. The specific features change as we ad- 
vance through the stages mentioned in the first principle. An example 
will illustrate the process in the case of single-variable data. 

In 1961 Yankee outfielder Roger Maris broke Babe Ruth's 
record of 60 home runs in a single season. Here is a back-to- 
back comparison of yearly home runs hit by Ruth (on the left) 
and by Maris during their years with the Yankees: 

















The overall shape of Ruth's distribution is roughly symmetric. The 
center is at about 46 home runs, in the sense that he hit more than 46 



half the time and fewer half the time. There are no strong deviations 
from the overall pattern. In particular, Ruth's famous 60 home runs in 
1927 do not stand out from the other values; it is Babe's best effort but 
not unusual in the context of his career. 

In contrast, Maris's record of 61 homers in 1961 is an outlier that 
falls clearly outside his overall pattern. That overall pattern (excluding 
the outlier) is again roughly symmetric and is centered at about 23. 
The different locations of the two distributions show Ruth's general 
superiority as a home-run hitter. 

To see the overall pattern of the distribution of a single variable, we 
learn to look for symmetry or skewness, for single or multiple peaks, 
for the center and the degree of spread about the center. Important 
deviations from a regular pattern include gaps and outliers. Notice that 
while constructing the display is an operation to be learned, interpreta- 
tion requires judgment. 

No distribution of real data has the perfect mirror symmetry of some 
mathematical shapes. Not all distributions are well described as either 
symmetric or skewed. Too much emphasis on classifying what we see 
will frustrate both teachers and students. Learn to observe marked fea- 
tures, not to debate unclear features. Note also that looking at data 
naturally leads to attempts to interpret what we see, as when we noticed 
that Ruth's 60 was not an unusual performance for him, while Maris's 
61 was an outstanding achievement far beyond his usual level. 

Interpreting the overall shape of a distribution is an important part 
of learning to look at data. The histogram in Figure 2 displays student- 
collected data on the lengths of words in Popular Science magazine. The 





0.20 r 



3 0.05 


3 4 

5 6 7 8 9 

10 11 12 13 14 15 

FIGURE 2. Student-collected data on the length of words in 
Popular Science magazine reveal a skewed distribution since 
shorter words are more common than longer ones. 


10 r 







380 400 420 440 460 480 500 520 540 

FIGURE 3. Data on the mean verbal SAT score by state reveal 
a double peak that reflects two different test-taking traditions: in 
some states most college-bound students take the SAT, whereas in 
other states only a few do since the majority take the ACT exam. 

distribution is right skewed because there are many two- to five-letter 
words and fewer long words. (The usual statistical terminology takes 
the direction of the skewness to be the direction of the longer tail, not 
the direction in which most observations are concentrated.) 

The histogram in Figure 3 shows the mean score by state on the verbal 
part of the Scholastic Aptitude Test (SAT). This distribution is double 
peaked. The peak near 425 represents states in which most college- 
bound students take the SAT; the higher-valued peak represents states 
in which most students take the American College Testing (ACT) exam- 
ination and only students applying to selective colleges take the SAT. 

Numerical Description 

Already in examining the Ruth and Maris home-run data we saw that 
calculation can help us describe data. By simple counting ("half more 
and half less") we can give numbers that make more exact the difference 
in centers that we see in the stemplots. The natural progression of math- 
ematical tools is expressed in the third organizing principle: graphics to 
numerical measures to mathematical models. 

In the case of the distribution of values of a single variable, the basic 
aspects to be described numerically are the center (or location) and the 
spread (or dispersion) of the distribution. (The older term "central ten- 
dency," which is both longer and less clear than "center" or "location," 
is rarely used by statisticians and should be abandoned.) There are two 
common sets of descriptive measures for location and spread: the me- 
dian with the quartiles (or perhaps other percentiles) and the mean with 



the standard deviation. Percentiles require only counting and an under- 
standing of simple fractions (1/4, 1/2, 3/4 for median and quartiles). 
The mean is the arithmetic average. So the mean, median, quartiles, 
and smallest and largest values can be introduced as students develop 
basic arithmetic skills. These simple measures form a helpful descrip- 
tive vocabulary. 

Experience with the connection between the shape of displayed data 
and numerical measures strengthens number sense. Although both the 
displays and the measures seem elementary, the amount of mathemati- 
cal understanding required to use them effectively (as opposed to simply 
calculating the measures) should not be underestimated. In one field test 
of new teaching material, for example, neither students nor the teacher 
could believe that adding observations to the right end of a particular 
distribution with many tied observations in the center left the median 
unchanged. 19 Hands-on experience with many sets of data, including 
attempts to estimate measures by looking at the display and discussing 
results, helps students construct their own understanding of such ap- 
parently simple operations as counting halfway up the ordered list (the 
median) and averaging all the values (the mean). 

Numerical description of a distribution by the median, quartiles, and 
extreme observations leads to a new graphic display, the boxplot. An 
example shows how useful this device can be. U.S. Department of 
Agriculture regulations group hot dogs into three types: beef, meat, and 
poultry. Do these types differ in the number of calories they contain? 
In Figure 4 three boxplots display the distribution of calories per hot dog 



E 160 


S 14 


9 120 






FIGURE 4. Three boxplots display visually the median, quar- 
tiles, and extremes of calories provided by various brands of hot 
dogs belonging to three standard types: beef, meat, poultry. One 
can easily see that poultry hot dogs as a group contain fewer 
calories per hot dog. 


among brands of the three types. The box ends mark the quartiles, the 
line within the box is the median, and the whiskers extend to the smallest 
and largest individual observations. We see that beef and meat hot dogs 
are similar but that poultry hot dogs as a group show considerably fewer 
calories per hot dog. 

Mathematical Models 

In this brief discussion of single-variable data, we have not yet men- 
tioned either the standard deviation or the final stage in the progression 
from graphical display to numerical description to mathematical model. 
The standard deviation has several disadvantages for data description. 
It is unpleasant to calculate with a basic calculator, is very sensitive to 
a few extreme values, and is difficult to motivate clearly. (The mean 
or median of the absolute deviations of the observations from their 
mean is preferable on all three counts.) 

Yet the standard deviation is very important in statistics, mainly be- 
cause it is the natural measure of spread for normal distributions. Nor- 
mal curves provide an example of a compact mathematical description 
of the overall pattern of a distribution of data. They are mathematical 
idealizations that do not catch the irregularity of real data or deviations 
such as outliers. Normal curves are, for example, perfectly symmetric. 

Most curriculum materials intended for general students stop short 
of presenting normal distributions. This is true, for example, of the 
Quantitative Literacy series 7 ' I0nl1 ' 15 developed jointly by the American 
Statistical Association and the National Council of Teachers of Math- 
ematics. One reason may be the traditional view of normal and other 
distributions as probability distributions, to be developed only after con- 
siderable study of probability. But it is not necessary to introduce for- 
mal probability to suggest that the heights of a large group of people of 
similar age and sex are roughly normal or that the stopping point of a 
spinner is roughly uniform over a circle. 

Figure 5 shows a histogram of the Iowa Test vocabulary scores of 
all 947 seventh-grade students in Gary, Indiana, with the normal curve 
that approximately describes the distribution of scores. It shows quite 
clearly how a normal curve provides an idealized mathematical model 
for certain distributions of data. 

Moving from particular observations to an idealized description of 
"all observations" is a substantial abstraction. The use of a mathe- 
matical model such as a normal or uniform distribution to formulate 
this abstraction is a substantial step toward understanding the power 
of mathematics. Computer simulation is quite helpful at this point. 







2 4 6 8 10 


FIGURE 5. A histogram of vocabulary scores of nearly 1000 
seventh-grade students shows close adherence to the idealized 
distribution of the bell-shaped normal curve. 

Students can formulate a "population model" on the basis of their ex- 
perience with data, enter their model into the computer, and simulate 
observations from the population. Comparing simulated data to the 
model provides more experience with probability and randomness. The 
basic properties of normal curves, the idea of standardizing observa- 
tions to the scale of standard deviation units about the mean, and the 
use of the standard normal table to calculate relative frequencies can be 
developed in the setting of models for regular patterns in data. 

Although distributions in the mathematical sense complete the pro- 
gression of descriptive methods for single-variable data, they must ap- 
pear rather late even when it is understood that distributions can appear 
before a full introduction to probability. Meanwhile, experience with 
several-variable data would have been advancing as students develop 
the necessary mathematical concepts and skills. The beginning study of 
two-variable data comes later than examination of a single variable, in 
accordance with our first principle, but usable mathematical models are 
more accessible in the two-variable case. 

The basic graph for two-variable data is the scatterplot, which pro- 
vides a setting for understanding coordinates in the plane. Clusters 
(female and male students?) and outliers in a scatterplot provoke dis- 
cussion. The simplest overall pattern is a linear trend. The mathe- 
matical model that gives a simple description of a linear pattern is a 
straight line with its equation. Numerical measures include measures 


of the center and spread of each variable separately, the slope of a fitted 
line as a description of linear relationship, and perhaps the correlation 
coefficient as a measure of the strength of linear association. 

The correlation coefficient, like the standard deviation, is tied to tra- 
ditional statistical models and methods whose advantages, while real, 
are not clear until a quite advanced stage of study. The correlation co- 
efficient is closely related to least squares regression; that is, correlation 
measures the strength of a specific kind of straight-line association. Just 
as the standard deviation should be delayed until normal distributions 
give it a context, correlation and least squares regression need not make 
their appearance until secondary school students undertake a substantial 
study of statistics for its own sake. 

Much of data analysis, while useful in its own right, can be taught 
from early elementary school through the first years of secondary school 
as part of the general effort to develop quantitative skills and reasoning. 
In this setting, straight lines can be fit by eye or by simple methods 
that are computationally easier than least squares and more resistant 
to extreme observations. The Quantitative Literacy material 10 offers a 
clear explanation of such methods for use in the middle grades. 

Other aspects of several variable data deserve priority over correla- 
tion and least squares regression. These include the distinction between 
explanatory and response variables, the relation of association to causa- 
tion, and the effects of unmeasured "lurking variables" on an observed 
association. These ideas are subtle but not computational; they are best 
grasped by guided experience with and discussion about actual data, us- 
ing a variety of display and computational methods; and they are closely 
related to an understanding of the kinds of explanations offered by the 
natural and social sciences. 

In teaching data analysis in a general school curriculum, topics should 
be chosen not for their importance in the discipline of statistics but for 
their immediate relevance to students, their usefulness in strengthening 
general quantitative understanding, and their contribution to develop- 
ing reasoning about uncertain data. Statistics is important in its own 
right more important than calculus in most occupations and that im- 
portance should be reflected in a substantial elective course in the upper 
secondary years that includes more advanced data analysis as well as 
data production, probability, and inference. 


Good data are as much a product of intelligent human effort as are 
compact disc players and hybrid corn. There are several reasons why 
producing data is an important part of teaching about data and chance. 


Data analysis is most effectively carried out on data with which we are 
intimately familiar, for familiarity suggests both expected features to 
look for and explanations for unexpected features. Statistical designs 
for producing data to answer specific questions are the conceptual bridge 
linking data analysis to classical probability-based inference. And there 
is no better cure for the extreme attitudes either unwarranted cynicism 
or misplaced trust with which statistical evidence is often greeted than 
experience that begins with a question and ends with answers based on 
data that we ourselves have produced. 

Data used in the teaching of statistics come from several sources. 
Much of it is provided data, numbers simply provided by the teacher 
or the text. With concerted effort to choose data on topics within stu- 
dents' experience or interests and to provide appropriate background 
information, provided data can offer a good setting for interpretation 
and discussion as well as for building skills. Provided data are more 
useful with older children who have the wider knowledge and experi- 
ence to understand the context of the data. Interesting information that 
students could not produce themselves can be put before the class, and 
the time and effort saved can be well used. Government data on nearby 
towns or neighborhoods, for example, often show patterns in popula- 
tion, housing, income, and health that are informative and surprising. 

A second category, class data, is collected in the classroom and is rel- 
evant primarily to students in the class without raising the question of 
whether conclusions about some larger population are warranted. Class 
data provide a natural setting for teaching data analysis, which has a 
similar restriction on the scope of its conclusions. Simple questions are 
a beginning: "How many children live in your house?" "How much 
money do you have in your pocket?" The first question produces whole 
number data, the second two-place decimals. Planning the production 
of data involves thinking ahead to the analysis that will be called for, a 
reminder as relevant to professionals armed with software as to teach- 
ers attentive to whether their students should face counts or decimals. 
Measurements can also produce class data: with a tape measure, find the 
shoulder width and armspan of all the students, then make a scatterplot 
and study the relationship revealed. 

Experiments are a third source of data. Experimentation is active 
data production. Observation, whether questioning or measuring, seeks 
to collect data without changing the people or things observed. In an 
experiment we actually apply some stimulus in order to observe the re- 
sponse. The distinction between explanatory and response variables 
an essential part of causal explanations is clearest in the setting of 
an experiment. The experiments most familiar in basic science, unlike 


the questions or measurements that produce class data, do invite con- 
clusions that apply to the world at large. When students heat a closed 
volume of air and watch a balloon expand, they are asked to understand 
not just the behavior of the one balloon but also the effect of heat on 
gases in general. This rather large conceptual leap is often left implicit. 
Moving from class data to statistically designed samples has the great 
advantage of making explicit the transition from data about this one 
class to data that represent a larger population. How to sample is a topic 
within statistics, with implications far broader than merely generating 
attractive data for analysis. Statistics also has much to say about how to 
experiment, although the advice is not relevant to most experiments in 
basic science. The design of samples and of experiments is a major topic 
in the systematic study of data production. But another topic conies 
first, both logically and in classroom experience: asking questions and 
measuring to produce class data both raise the issue of measurement. 


To measure a characteristic means to represent it by a number. This 
basic notion already introduces an abstraction. Thinking about mea- 
surement leads at length to a mature grasp of why some numbers are 
informative and others are irrelevant or nonsensical. First, what is a 
valid (appropriate or meaningful) way to measure a particular charac- 
teristic? Begin with tangible physical characteristics. Length is easy we 
agree that a ruler will do it. Area is harder, because we have no device 
that we can "put beside" the many shapes possible in two dimensions 
as we put a ruler beside any length. We must concern ourselves with 
understanding the characteristic to be measured, with devising a satis- 
factory instrument, and with the units that result and their relations to 
other units. Even for physical measurements the study of these ques- 
tions extends throughout the school years both in mathematics and in 

But the validity of physical measurements is simple compared with 
the measurement problems of the social and behavioral sciences. What 
is a good way to measure how rich a family is or the friendliness of a 
fellow student? What do the Iowa Tests or the ACT and SAT college 
entrance examinations really measure? A detailed examination of such 
questions would lead too far afield. But students should be encouraged 
always to ask whether data are in fact valid for the proposed use. Drivers 
over 65 years of age are involved in more fatal accidents than drivers 
aged 16 and 17. So teens aren't so risky after all? No there are many 
more drivers over 65. The rate rather than the count of accidents is the 


appropriate measurement, and the fatal accident rate for teens is about 
three times that for the elderly. 

The second major aspect of the quality of measurements, after va- 
lidity, is accuracy. A measuring process may show systematic error, or 
bias, as when a scale always reads 3 pounds low. Bias is a straightfor- 
ward idea only when the "true value" that measurement should yield is 
clearly understood. Possible bias in SAT scores is a continuing source 
of intense debate, since no "correct" value is available for comparison. 
As usual, physical measurement is much more straightforward than be- 
havioral or social measurement. 

A measuring process also shows variation; that is, repeated measure- 
ments of the same quantity do not give identical results. The variations 
in common instruments such as bathroom scales and tape measures are 
small relative to the desired accuracy, so we are accustomed to ignoring 
variation in measurement. Activities that demonstrate measurement 
variation are needed. Requiring students to interpolate between scale 
markings when measuring length or weight, or to estimate a length or 
count by eye, provides a set of varying measurements whose distribu- 
tion can be displayed and discussed with the tools of data analysis. Bias 
is described by the center of the distribution of measurements and vari- 
ation by the spread. 

Measurement activities followed by discussion of the data they pro- 
duce increase students' sensitivity to the issue of the quality of mea- 
surements. Here is an example from a college class. 

The instructor asked each student to measure and record his 
or her pulse rate (heartbeats per minute) on a piece of paper. 
A stemplot of the collected data showed an outlier that almost 
certainly resulted from a gross error, though no one would ad- 
mit having recorded a seated pulse rate of 1 80. The stemplot 
also showed a suspicious concentration of pulse rates ending in 
0. Questioning revealed that several students had learned in 
aerobics classes to count beats for 6 seconds and multiply by 
1 0. This led to a discussion of the measurement methods used. 
Most students had counted beats for 60 seconds. The class de- 
cided that this is more accurate than the aerobics class method, 
but it suffers from partial beats at the beginning and end of the 
60-second period. Someone suggested timing exactly 50 beats 
with a stopwatch and calculating beats per minute from this 
time. This was accepted as a more accurate practical measure- 
ment method. 


Design of sample surveys and experiments is a core topic in statistics 
and a major transition in concepts. Data analysis emphasizes under- 
standing the specific data at hand. Now the data are regarded as repre- 
senting a larger population. It is the population we seek to understand. 
Students do not find this added abstraction easy to assimilate. They 
persist, for example, in trying to explain variable results when an ex- 
perimental task is carried out by several students in terms of individual 
characteristics of Sarah, Matthew, and Ruth. The "sampling" point of 
view regards these students as representative of a large population of 
students. We are no longer interested in individual features that may 
explain the performance of Sarah, Matthew, and Ruth. 

The transition from data analysis to inference follows a parallel path 
in mathematical abstraction. The sample mean x is no longer just a 
single number, a measure of location for these data. It is a realization 
of a random variable to be considered against the background of the 
distribution of the random variable; it must be viewed against what 
would happen if we repeated the data production process many times. 
The difficulty of these new ideas cannot be disguised. 

Fortunately, the intimate connection of designed data production 
with the ideas of probability and the logic of inference need not ap- 
pear at once. There is much valuable insight into data to be gained 
first. It is very important, for example, to recognize unrepresentative 
data. Anecdotal evidence based on a few individual cases known to 
us influences our thinking in ways that cannot withstand examination 
and therefore must be examined. Individual cases catch our attention 
because they are unusual in some way or because they occur in our im- 
mediate environment. Examples and discussion will show that there is 
no reason to expect these cases to be in any way typical. 

Improper sampling methods, especially voluntary response samples 
in which the respondents choose themselves, are also fair game. Here 
is an example: 

Advice columnist Ann Landers conducts a voluntary response 
survey every few years by asking her readers to respond to a 
provocative question. The results are always good for news ar- 
ticles and radio interviews that publicize her column. Her first 
survey is the most instructive because a comparison is available. 
In 1975 Ann Landers asked "If you had it to do over again, 
would you have children?" Almost 70% of the nearly 10,000 
respondents said "no." Many accompanied their responses by 
heart-rending tales of the cruelties inflicted on them by their 


children. It is the nature of voluntary response to attract peo- 
ple with strong feelings, especially negative feelings, about the 
issue in question. A nationwide random sample commissioned 
in reaction to the attention paid to Ann Landers's results found 
that 91% of parents would have children again. 

Voluntary response can easily produce 70% "no" when the truth is 
90% "yes." Such data carry no useful information about anyone except 
the people who stepped forward. Yet the news media not only report 
voluntary response data as if they described a general population, they 
also operate call-in and write-in polls that produce more such data. Alert 
students will easily find examples. Discussion of anecdotal evidence 
and voluntary response makes clear the need for a systematic method 
for selecting samples. 

The statistician's recommended method is to let impersonal chance 
select the sample. Random sampling eliminates the biases of personal 
choice, whether by the sampler or by the respondents. The deliberate 
use of chance is the most important statistical principle for produc- 
ing data. It seems at first unnatural to abandon human judgment, but 
chance appears less outrageous when set against anecdotal evidence and 
voluntary response. The use of chance is illustrated by simple random 
samples, which give all possible samples of the stated size the same 
chance to be the sample actually chosen. 

Simple random samples are easy to experience in the classroom, first 
by drawing names from a hat or varicolored beads from a sampling 
bowl. Use of a random number table follows, and finally computer 
simulation. Do recall the warning that too rapid introduction of the 
computer will obscure the nature of random selection. The more elabo- 
rate random sampling designs used in national sample surveys need not 
appear in introductory instruction. 

The simplest randomized comparative experiments are closely related 
to simple random samples. Once again the need for good design can 
be made apparent by discussion of some uncontrolled or unrandomized 
experiments. Here is an example: 

A political scientist interested in the effectiveness of propa- 
ganda in changing opinions conducted an experiment with stu- 
dent subjects. The students took a test of their attitude toward 
Germany, then read German propaganda regularly for several 
months, after which their attitude was again measured. The year 
was 1 940. Between test and retest, Germany invaded and con- 
quered Holland and France. The students' attitude toward Ger- 
many changed drastically, but we shall never know how much 
of this change was due to reading German propaganda. 


The design of this experiment had a form familiar in laboratory ex- 
periments in the natural sciences: 

Observation ** Treatment >- Observation 

Outside the controlled environment of the laboratory, experiments with 
such simple designs often fail to yield useful data. The effect of the 
treatment cannot be distinguished from the effect of external variables, 
though not all such disturbances are as dramatic as the fall of France. 

Statistically designed experiments involve two basic principles: com- 
parison (or control) and randomization. The simplest randomized com- 
parative design compares two treatments, one of which may simply be 
a control treatment such as not reading propaganda. Here is the design 
in outline: 

Group 1 > Treatment 1 > Observation 

Allocation ^ 

Group 2 > Treatment 2 > Observation 

The random allocation assigns a simple random sample of the subjects 
to Treatment 1; the remaining subjects receive Treatment 2. Random- 
ization assures that there is no bias in assigning subjects to treatments. 
The groups are therefore similar (on the average) before the treatments 
are imposed. Comparison assures that outside forces act equally on 
both groups. If care is taken to treat all subjects similarly except for the 
experimental treatments, any systematic difference in response must re- 
flect the effect of the treatments. The logic of comparative randomized 
experiments allows conclusions about causation the response is not 
just associated with the treatment but is actually caused by it. 

As in the case of sampling, more elaborate designs are common in 
practice but need not appear in beginning instruction. Classroom expe- 
rience with randomization is easy and valuable. Consider, for example, 
tokens such as gumdrop figures that represent subjects to be assigned to 
two competing treatments for severe headaches. Students carry out the 
random assignment. Some of the tokens bear a mark on the bottom, 
invisible when the randomization is done. These subjects, unknown to 
the experimenters, have a brain tumor that will render any treatment in- 
effective. How evenly did randomization divide these subjects between 
the two groups? Do the randomization repeatedly and display the dis- 
tribution of counts. Repeated randomization provides experience with 
random variation that leads toward probability and inference. 


Some Cautions 

With the fundamentals of both data analysis and data production in 
hand, older students can contemplate serious statistical studies. Ex- 
amples from recent curriculum projects include a sample of student 
opinion about the selections served in the school cafeteria; a sample of 
vehicles at a local intersection, classified by type and home county as 
revealed by the license plate; and an experiment on the effect of dis- 
tance and angle on success in shooting a nerf basketball. The design of 
such studies provides valuable experience in applying statistical ideas. 
Analysis of real data to arrive at solid conclusions is satisfying. But the 
practical problems of producing the data must be anticipated and kept 
within acceptable limits. 

Here is an excerpt from a report of a careful study 1 of new statistics 
material for secondary schools. Some of the data production activities 
were quite elaborate, including both the road traffic survey and the nerf 
basketball experiment. Their experience is cautionary. 

Our field test experiences have convinced us that data collection is an important 
component of statistics education for at least two reasons. First, learning how 
to design and conduct data collection activities (e.g., determining independent 
and dependent variables and sample size) is fundamental to statistics. Second, 
data collection is a motivating experience that makes statistical analysis more 
meaningful and interesting to students. 

Our experiences also convinced us, however, that data collection can present 
some formidable challenges in the classroom. For example, our field test teach- 
ers report that they spent an inordinate amount of class time collecting data as 
opposed to exploring and analyzing data, only to find that students' data was in- 
complete or inaccurate. These challenges proved to be so disruptive to academic 
progress that the teachers grew reluctant to conduct statistical investigations that 
depended on data collection. 


Chance variation can be investigated empirically, applying the tools of 
data analysis to display the regularity in random outcomes. Probability 
gives a body of mathematics that describes chance in much more detail 
than observation can hope to discover. Probability theory is an impres- 
sive demonstration of the power of mathematics to deduce extensive 
and unexpected results from simple assumptions. 

Coin tossing, for example, is described simply as a sequence of in- 
dependent trials each yielding a head with probability 1/2. From this 
unassuming foundation follow such beautiful results as the law of the 
iterated logarithm, which gives a precise boundary for the fluctuations 
in the count of heads as tossing continues. The distribution of the count 


50 - 



20 30 




FIGURE 6. The law of the iterated logarithm describes the region 
of fluctuations in coin tossing: the center line is the mean /2, 
bounded on either side by curves whose distance from the center 
line is 

of heads after n tosses of a fair coin has a mean of n/2 9 which when 
plotted against n appears as a straight line (see Figure 6). The standard 
deviation of the count of heads in n tosses is O.Sx/w. The law of the 
iterated logarithm says that fluctuations in the count of heads extend 
y^loglogfl standard deviations on either side of the mean. The count 
of heads plotted against n will approach within any given distance of 
this boundary infinitely often as tossing continues, but will cross it only 
finitely often. Data analysis, even aided by computer simulation, could 
never discover the law of the iterated logarithm. 

As with other beautiful and useful areas of mathematics, probabil- 
ity has in practice only a limited place in even secondary school in- 
struction. Because the fundamentals of probability are mathematically 
rather simple, it is easy to overlook the extent to which the concepts of 
probability conflict with intuitive ideas that are firmly set and difficult 
to dislodge by the time students reach secondary school. Misconcep- 
tions often persist even when students can answer typical test questions 
correctly. The conceptual difficulty of probability ideas is affirmed by 
both the experience of teachers and by research. 5 ' 21 


Guided experience with randomness in earlier years is an important 
prerequisite to successful teaching of formal probability. It is no ac- 
cident that mathematical probability originated in the study of games 
of chance, one of the few settings in which simple random phenomena 
are observed often enough to display clear long-term patterns. Teach- 
ing can attempt to recapitulate this historical development by recording 
data from chance devices and later from random sampling and com- 
puter simulations. But no matter whether such experience occurs early 
or late in a student's development, it takes significant time to gain ap- 
propriate insight into the behavior of random events. 


The first steps toward mathematical probability take place in the con- 
text of data from chance devices in the early grades. Learn to look at the 
overall pattern and not attempt a causal explanation of each outcome 
("She didn't push the spinner very hard"). This abstraction is made 
easier because looking for the overall pattern of data is one of the core 
strategies of data analysis. 

Next recognize that, although counts of outcomes increase with added 
trials, the proportions (or relative frequencies) of trials on which each 
outcome occurs stabilize in the long run. Probabilities are the mathe- 
matical idealization of these stable long-term relative frequencies. As 
students learn the mathematics of proportions, study of probability can 
begin with assignments of probabilities to finite sets of outcomes and 
comparison of observed proportions to these probabilities. 

Comparison of outcomes to probabilities can be frustrating if not 
carefully planned. Computer simulation is very helpful in providing 
the large number of trials required if observed relative frequencies are 
to be reliably close to probabilities. In short sequences of trials, the 
deviations of observed results from probabilities will often seem large 
to students. Psychologists 20 have noted our tendency to believe that the 
regularity described by probability applies even to short sequences of 
random outcomes. This belief in an incorrect "law of small numbers'" 
explains the behavior of gamblers who see a run of winning throws with 
dice as evidence that the player is "hot," a causal explanation offered 
because we greatly underestimate the probability of runs in random 

Ask several people to write down a sequence of heads and 
tails that imitates 10 tosses of a balanced coin. How long was 
the longest run of consecutive heads or consecutive tails? Most 
people will write a sequence with no runs of more than two 
consecutive heads or tails. But in fact the probability of a run 


of three or more heads In 10 independent tosses of a fair coin is 
0.508, and the probability of either a run of at least three heads 
or a run of at least three tails is greater than 0.8. 

Probability calculations involving runs are quite difficult this is a 
good area for computer simulation. The runs of consecutive heads or 
consecutive tails that appear in real coin tossing (and are predicted by 
probability theory) seem surprising to us. Since we don't expect to see 
long runs, we may conclude that the coin tosses are not independent or 
that some influence is disturbing the random behavior of the coin. 

The same misconception appears on the basketball court. If a player 
makes several consecutive shots, both fans and teammates believe that 
he or she has a "hot hand" and is more likely to make the next shot. 
Yet examination of shooting data 22 shows that runs of baskets made or 
missed are no more frequent than would be expected in a sequence of 
independent random trials. Shooting a basketball is like throwing dice, 
though of course the probability of making a shot varies from player 
to player. As these examples suggest, even the idea of probability as 
long-term relative frequency is quite sophisticated and needs careful 
empirical backing. 

Somewhat later a thorough understanding of proportions motivates 
the mathematical model for probability: a sample space (set of all pos- 
sible outcomes) and an assignment of probability satisfying a few basic 
laws or axioms that include the addition rule P(A or B] = P(A) + P(B) 
for disjoint events. Further additive laws for simple combinations of 
events can be derived from these or, more simply, motivated directly 
from the behavior of proportions. These additive laws are the mathe- 
matical content of elementary probability. 

At this point in the development of mathematical probability, let 
us pause for some nonnumerical exercises that apply probability laws 
along with another aspect of mathematical thinking that is not natu- 
ral in students: careful and literal reading of logical statements. Psy- 
chologists studying probability concepts offer many exercises that reveal 
misconceptions and can help to correct them. For example, Tversky 
and Kahneman 21 presented college students with a personality sketch 
of a young woman and then asked which of these statements was more 

* Linda is a bank teller. 

Linda is a bank teller and is active in the feminist movement. 

About 85% of the students chose the second statement, even though 
this event is a subset of the first. This error persisted despite various 
attempts at alternative presentations that might make the issue more 
transparent. The subjects had not studied probability. "Only" 36% of 


social science graduate students with several statistics courses to their 
credit gave the wrong answer in a similar trial. There is thus some 
hope that study helps us recognize the relevance of mathematical facts 
about probability in everyday thinking. Nisbett et al. 17 report before- 
and-after comparisons that provide stronger evidence of the effect of 
formal study. Emphasis on the conceptual and qualitative aspects of 
probabilistic thinking, both prior to and in company with study of the 
mathematics of probability, is most worthwhile. 

Further Study 

The development of substantial applicable skills, as opposed to a basic 
conceptual grasp of probability, requires more detailed study. At this 
point we leave the core domain of mathematical concepts to which all 
students should be exposed. There are several logical paths into inter- 
mediate probability. The choice of material will depend, for example, 
on whether probability will be pursued as an important topic in its own 
right or whether it is intended primarily to lead to statistical inference. 

First, a negative recommendation: do not dwell on combinatorial 
methods for calculating probabilities in finite sample spaces. Combi- 
natorics is a different and harder subject than probability. Students 
at all levels find combinatorial problems confusing and difficult. The 
study of combinatorics does not advance a conceptual understanding of 
chance and yields less return than other topics in developing the ability 
to use probability modeling. In most cases all but the simplest counting 
problems should be avoided. 

A more fruitful step forward from the basics of probability is to con- 
sider conditional probability, independence, and multiplication rules. 
Knowledge of the occurrence of an event A often modifies the probabil- 
ity assigned to another event B. For example, knowing that a randomly 
selected university professor is female reduces the probability that the 
professor's field is mathematics. The conditional probability of B given 
A, denoted by P(B\A\ need not be equal to P(B)\ if the two are equal, 
events A and B are independent. These notions involve both new ideas 
and basic skills that are invaluable in constructing probability models 
in the natural and social sciences. 

It is quite possible to present the idea of independence and the mul- 
tiplication rule P(A and B) = P(A)P(B) for independent events with 
little if any attention to conditional probability in general. This path 
is attractive if the goal is to reach statistical inference most efficiently 
and also avoids the considerable conceptual difficulties associated with 
conditional probability. The binomial distributions for the count of suc- 
cesses in a fixed number of independent trials are quickly within reach, 


as are other interesting applications such as reliability of complex sys- 

If conditional probability is avoided, stress the qualitative meaning of 
independence and the danger of casually assuming that independence 
holds. The essay by Kniskal 9 contains examples and reflections on the 
casual assumption of independence, with emphasis on "independent" 
testimony to alleged miracles. Topics related to independence, to bino- 
mial distributions, and to the multiplication rule for independent events 
should be staples of upper-grade secondary mathematics. 

A careful study of conditional probability is attractive when the goal 
is to enable students to construct and use mathematical descriptions 
of processes at a moderately advanced level. Modeling of multistage 
processes that are not deterministic requires conditional probabilities. 
To give only a single example, the issue of false positives in testing for 
rare conditions applies conditional probability to questions as current 
as testing for drugs, the use of lie detectors, and screening for AIDS 
antibodies. Here is an example based on a recent report, 6 where a 
detailed statistical analysis can be found: 

The ELISA test was introduced in the mid-1980s to screen 
donated blood for the presence of AIDS antibodies. When an- 
tibodies are present, ELISA is positive with a probability of 
about 0.98; when the blood tested is not contaminated with an- 
tibodies, the test gives a positive result with a probability of 
about 0.07. These numbers are conditional probabilities. If one 
in a thousand of the units of blood screened by ELISA contain 
AIDS antibodies, then 98.6% of all positive responses will be 
false positives. 

The calculation of the prevalence of false positives among ELISA 
blood screening tests for AIDS antibodies can be carried out with a 
simple tree diagram such as that displayed in Figure 7. Students armed 
with an understanding of conditional probability and tree diagrams can 
easily program computer simulation of processes too complex to study 

Conditional probability brings a new set of conceptual difficulties that, 
like those in the early study of probability, can be easily and unwisely 
overlooked if instruction is overly directed at teaching definitions and 
rules. Students find the distinctions among P(A\B) 9 P(B\A), and P(A 
and B) hard to grasp. Display a photograph of an attractive and well- 
dressed woman and ask the probability that she is a fashion model. The 
answers show that the question is interpreted as asking the conditional 
probability that a woman known to be a fashion model is attractive 
and well dressed. That is, respondents confuse P(A\B) and P(B\A). 



Unit of 




.98 .^ ELISA .00098 

^ ' ELSSA .00002 





P (ELISA +) 


.00098 + .06993 




FIGURE 7. The calculation of false positives in the ELISA test for AIDS antibodies can 
be carried out in a tree diagram in which the appropriate conditional probabilities are 
multiplied along each branch. 

Qualitative exercises in identifying the information A that is known and 
the event B whose probability is wanted are an essential preliminary to 
formal work with P(B\A). 

Transition to Inference 

Random sampling and experimental randomization provide experi- 
ence with randomness that motivates not only the study of probability 


but also the reasoning of probability-based inference. Repeated sam- 
pling or repeated experimental randomization clearly produces variable 
results. This variation is random in the technical sense, rather than 
haphazard, because the design uses an explicit chance mechanism. So 
an opinion poll's conclusion that 61% of all American adults want a 
national health insurance system requires a margin of error that re- 
flects the probable degree of random variation in similar sample surveys. 
Similarly, the conclusion that a new medical treatment outperforms a 
standard treatment can be sustained only if the margin of superiority 
exceeds the probable random variation in similar experiments. 

The random outcomes observed from data production are statistics 
such as sample proportions and sample means. Sample statistics are 
random variables (random phenomena having numerical values). The 
regular long-term behavior of such statistics in repeated sampling or 
repeated experimental randomization is described by a sampling dis- 
tribution. It is usual to view sampling distributions as probability dis- 
tributions of random variables. Random variables, their distributions, 
and their moments make up another block of material in intermediate 

Proportions involve the distribution of a count, which is binomial 
under slightly idealized assumptions. Sample means have a normal dis- 
tribution if the population distribution is normal. General rules for 
manipulating means and variances of random variables apply to sam- 
ple proportions and means. In particular, the standard deviations of 
sample proportions and means both decrease at the rate \l\Jn as the 
sample size n increases, a fact that leads to an understanding of the 
advantages of larger samples. 

What happens as the number n of observations grows without bound? 
The major limit theorems of probability address this question. The law 
of large numbers says that sample proportions and means approach (in 
various senses) the corresponding proportions and means in the under- 
lying population. The central limit theorem says that both proportions 
and means become approximately normally distributed as the sample 
size grows. 

Figure 8 illustrates the central limit theorem in graphical form. It 
begins with the distribution of a single observation that is right skewed 
and far from normal. Distributions of this form are often used to de- 
scribe the lifetime in service of parts that do not wear out. The mean of 
this particular distribution is 1 . The other curves in the figure show the 
distribution of the mean 3? of samples of size 2 and of size 10 drawn 
from the original distribution. The characteristic normal shape is al- 
ready starting to emerge when only 10 observations are averaged. A 
computer simulation could show the effect even more dramatically. 



n = 10 


FIGURE 8. The central limit theorem 
in action: the distribution of means of 
samples drawn from a skewed distribu- 
tion (a) displays a progression toward the 
normal distribution as the sample size 
increases from 2 (b) to 10 (c). 

This is a substantial body of material that is quite forbidding if for- 
mally presented. Traditional college instruction in statistics insists that a 
substantial dose of probability at least topics on independence and on 
random variables precede the study of inference. Some understanding 
of independence and of distributions with their means and standard de- 
viations is certainly needed. But the degree of mathematical formalism 
with which these topics are traditionally taught is generally unnecessary 
at the college level and out of the question in secondary school. Both 
the length and the difficulty of the path to statistics via formal proba- 
bility argue against this traditional approach. As Garfield and Ahlgren 
conclude, 5 

. . . Teaching a conceptual grasp of probability still appears to be a very difficult 
task, fraught with ambiguity and illusion. Accordingly, we make the pragmatic 
recommendation for two research efforts that would proceed in parallel: one 
that continues to explore means to induce valid conceptions of probability, 
and one that explores how useful ideas of statistical inference can be taught 
independently of technically correct probability. 

Fortunately, the empirical emphasis of data analysis, developed grad- 
ually beginning in the early grades, offers a setting for teaching both ba- 
sic probability and elementary inference. Simulation, first physical and 
then using software, can demonstrate the essential concepts of probabil- 
ity and is particularly suited to displaying sampling distributions. Only 
quite informal probability is needed to think about sampling distribu- 
tions. As the earlier discussion of normal distributions indicated, data 
description provides an adequate context for distributions as idealized 
mathematical models for variation. The core mathematics curriculum 
taught to all students should include data analysis and an empirical 
introduction to only basic probability concepts and laws at about the 
level of the Quantitative Literacy material. 15 



Statistics is concerned with the gathering, organization, and analysis 
of data and with inferences from data to the underlying reality. "Infer- 
ence from data to reality" is a knotty topic indeed, with much room for 
disagreements of a philosophical nature. It is not surprising that statis- 
ticians disagree on the most fruitful approach to inference. Barnett 2 
gives a comparative overview of the competing positions. 

Bayesian or Classical? 

The most important philosophical divide separates Bayesian infer- 
ence from classical inference. Some understanding of the distinction 
is essential to wise curriculum decisions. The question of inference in 
simplest form is how to draw conclusions about a population parameter 
on the basis of statistics calculated from a sample. A parameter is a 
number that describes the population, such as the mean height ju of all 
American women age 18 to 22. A relevant statistic in this case is the 
sample mean height "x of a random sample of young women. For pur- 
poses of inference we imagine how x would vary in repeated samples 
from the same population. The sampling distribution of the statistic 
describes this variation. The sampling distribution reflects the underly- 
ing parameter in this case p, is the mean of the distribution of x. It is 
because the sampling distribution depends on the unknown parameter 
that the statistic carries information about the parameter. 

Classical inference is rooted in the concept of probability as long-term 
regularity and in the corresponding idea that the conclusions of infer- 
ence are expressed in terms of what would happen in repeated data 
production. To say that we are "95% confident that // lies between 64.5 
and 64.7 inches" is shorthand for "We got this interval by a method 
that is correct in 95% of all possible samples." Probability statements 
in classical inference apply to the method rather than to the specific con- 
clusion at hand indeed, probability statements about a specific conclu- 
sion make no sense because the population parameter is fixed, though 

The Bayesian approach wishes to bring prior information about the 
value of the parameter into play. This is done by regarding the 
parameter ju as a random quantity with a known probability distribu- 
tion that expresses our imprecise information about its value. The mean 
height IL of all American young women is not random in the traditional 
sense. But it is uncertain. I am quite sure that JJL lies between 54 inches 
and 72 inches, and I think it more likely that // lies near the center of 
this range. My subjective assessment of uncertainty can be expressed in 
a probability distribution for /u. 


In the Bayesian view the concept of probability is expanded to include 
such personal or subjective probabilities. What is new here is not the 
mathematics, which remains the same, but the interpretation of proba- 
bility as representing a subjective assessment of uncertainty rather than 
a long-run relative frequency. The sampling distribution of the statistic 
x is now understood to state conditional probabilities of the values of x 
given a value for /z. A calculation then combines the prior information 
with the observed data to obtain the conditional distribution of JLI given 
the data. (The discrete form of this calculation uses a simple result 
about conditional probabilities known as Bayes' theorem, from which 
the Bayesian school takes its name.) The conclusions of inference are 
expressed in terms of probability statements about the unknown param- 
eter itself: the probability is 95% that the true mean lies between 64.5 
and 64.7 inches. 

The Bayesian conclusion is certainly easier to grasp than the classical 
statement. Moreover, prior information is important in many problems. 
Statisticians generally agree that Bayesian methods should be used when 
the prior probability distribution of the parameter is known. What is 
disputed is whether usable prior distributions are always available, as 
Bayesians contend. Non-Bayesian statisticians do not think that my sub- 
jective assessment is always useful information and so are not willing to 
make general use of subjective prior distributions. The apparently clear 
conclusion of a Bayesian analysis can depend strongly on assumptions 
about the prior distribution that cannot be checked from the data. 

For introductory instruction about inference, Bayesian methods have 
several disadvantages. They require a firm grasp of conditional prob- 
ability. Indeed, students must understand the distinction between the 
conditional distribution of the statistic given the parameter and the con- 
ditional distribution of the parameter given the actually observed value 
of the statistic. This is fatally subtle. The subjective interpretation of 
probability is quite natural, but it diverts attention from randomness 
and chance as observed phenomena in the world whose patterns can 
be described mathematically. An understanding of the behavior of ran- 
dom phenomena is an important goal of teaching about data and chance; 
probability understood as personal assessment of uncertainty is at best 
irrelevant to achieving this goal. The line from data analysis through 
randomized designs for data production and probability to inference is 
clearer when classical inference is the goal. 

Two types of inference, confidence intervals and significance tests, fig- 
ure in introductory instruction in classical statistical inference. The rea- 
soning behind both types of inference can be introduced informally in 
discussions about data. Formal treatment and specific methods should 


be reserved for upper-grade secondary courses in probability and statis- 
tics, and no attempt should be made to present more than a few specific 
procedures. Particularly in the case of significance tests, a formal ap- 
proach obscures the reasoning to such an extent that it may be better to 
avoid hypotheses and test statistics altogether. 

Confidence Intervals 

The reasoning behind confidence statements is relatively straightfor- 
ward. What is more, news reports of opinion polls and their margins 
of error provide a steady supply of examples for discussion. How is it 
that a sample of only 1 500 people can accurately represent the opinion 
of 185 million American adults? Random sampling provides a part of 
the explanation; sampling distributions provide the rest, and confidence 
intervals explain what the margin of error means. 

Confidence statements can be introduced after students have some 
experience with simulation of sampling distributions. The distinction 
between population and sample, the idea of random sampling, and the 
notion of a sampling distribution are fundamental to inference. Sim- 
ulation allows the gradual introduction of confidence intervals during 
the exploration of sampling and sampling distributions. The ideas of 
confidence intervals can be taught via graphical display of simulated 
samples. 10 A more formal approach requires familiarity with normal 

Suppose that in a large county 30% of high school students drive cars 
to school. Asking a simple random sample of 250 students whether they 
drove to school today produces 250 independent observations, each with 
probability 0.3 of being "yes." The proportion p of "yes" responses in 
the sample varies from sample to sample. Simulate (say) 1000 samples 
and display the sampling distribution of p. It is approximately normal, 
with mean 0.3 and standard deviation 0.029. Repeated simulations 
of samples of various sizes from this population demonstrate that the 
center of the sampling distribution remains at 0.3 and that the spread 
is controlled by the size of the sample. In large samples (about 1000 or 
so) the values of the sample statistic p are tightly concentrated around 
the population parameter p = 0.3. Students can see empirically that 
samples of this size allow good guesses about the entire population. 

But just how good are guesses based on a sample? We can quantify the 
answer by describing how the statistic p varies in repeated sampling. It is 
a basic fact of normal distributions that about 95% of all observations 
lie within two standard deviations on either side of the mean. So in 
repeated sampling, 95% of all samples of 250 students give a sample 


proportion p within about 0.06 of the true proportion 0.3 who drive to 
school. The simulation shows that this is so. 

Now suppose that a sample of 250 students in another large county 
finds 105 who drive to school. We guess that the true proportion p of 
all students in this county who drive to school is close to p = 105/250 = 
0.42. If (as is true) the variability is about the same as in the county 
we simulated, p lies within 0.06 of p in 95% of all samples. We say 
we are 95% confident that the unknown population proportion p lies in 
the interval 0.42 0.06. More generally, the interval p 0.06 is a 95% 
confidence interval for the unknown p. 

Figure 9 illustrates the behavior of a confidence interval in repeated 
samples. As repeated samples of size 250 are drawn, some of the inter- 
vals p 0.06 cover the true proportion of/?, while others do not. But in 
the long run, 95% of all samples produce an interval covering the true 
p. That is, the probability that the random interval p 0.06 contains 
p is 0.95. As is generally the case in classical inference, this probability 
refers to the performance of the method in an indefinitely large number 
of repeated samples. 

The first portion of the argument above belongs to the study of sam- 
pling and simulation and is essentially an empirical demonstration of 
the surprising trustworthiness of samples that seem small relative to 
the size of the population. The facts that emerge from such sampling 
demonstrations are much more important than the formal dress we give 
them in the second stage of the argument. The second stage belongs to 
a more advanced study of inference. The qualitative conclusion that 
most sample results lie close to the truth is made quantitative by giving 
an interval and a level of confidence. The nature of this conclusion and 
its limitations both need emphasis. 

What are the grounds of our confidence statement? There are only 
two possibilities. 

1. The interval 0.42 0.06 contains the true population proportion 

2. Our simple random sample was one of the few samples for which 
p is not within 0.06 points of the true p. Only 5% of all samples 
give such inaccurate results. 

We cannot know whether our sample is one of the 95% for which the 
interval catches p or one of the unlucky 5%. The statement that we are 
95% confident that the unknown p lies in 0.42 0.06 is shorthand for 
"We got these numbers by a method that gives correct results 95% of 
the time." 

As for the limitations on this reasoning, remember that the margin 
of error in a confidence interval includes only random sampling error. 




FIGURE 9. The behavior of a confidence interval in repeated 
samples from the same population. The normal curve is the 
sampling distribution of the sample proportion p centered at the 
population proportion p. The dots are the values of p from 25 
samples, with the confidence interval indicated by arrows on either 
side. In the long run 95% of these intervals will contain p. 

In practice there are other sources of error that are not accounted for. 
For example, national opinion polls are usually conducted by telephone 
using equipment that dials residential telephone numbers at random. 
Telephone surveys omit households without phones. Moreover, poll- 
sters often find that as many as 70% of the persons who answer the 
phone are women. Men will be underrepresented in the sample un- 
less steps are taken to contact males. These facts of real statistical life 
introduce some bias into opinion polls and other sample surveys. 

Significance Tests 

The purpose of a confidence interval is to estimate a population pa- 
rameter and to accompany the estimate with an indication of the un- 
certainty due to chance variation in the data. Significance tests do not 


provide an estimate of an unknown parameter, but only an assessment 
of whether an effect or difference is present in the population. The mere 
recognition that such an assessment is needed, that not all observed out- 
comes signify a real underlying cause, already shows statistical sophis- 
tication. Judges of science fair displays who talk to the able students 
who have prepared them find that any effect in the desired direction, 
however small, is regarded as convincing. The role of chance variation 
is not recognized. 

Statistical significance is a way of answering the question "Is the ob- 
served effect larger than can reasonably be attributed to chance alone?" 
Here is the reasoning of significance tests presented informally in the 
setting of an important example: 

During the Vietnam era, Congress decided that young men 
should be chosen at random for service in the army. The first 
draft lottery was held in 1970. Birth dates were drawn in ran- 
dom order and men were drafted in the order in which their 
birth dates were selected. After the drawing, news organiza- 
tions claimed that men born late in the year were more likely 
to get low draft numbers and so to be inducted. Data analysis 
(Figure 10) does suggest an association between birth date and 
draft number. A statistic that measures the strength of the as- 
sociation between draft number (1 to 366) and birth date (I to 
366 beginning with January 1) is the correlation coefficient. In 
fact, r = -0.226 for the 1970 lottery. Is this good evidence that 
the lottery was not truly random? 

A significance test approaches the issue by asking a probability ques- 
tion: Suppose for the sake of argument that the lottery were truly ran- 
dom; what is the probability that a random lottery would produce an r 
at least as far from as the observed r = -0.226? Answer: The proba- 
bility that a random lottery will produce an r this far from is less than 
0.001. Conclusion: Since an r as far from as that observed in 1970 
would almost never occur in a random lottery, we have strong evidence 
that the 1970 lottery was not random. 

Figure 10 displays the scatterplot of draft numbers assigned to each 
birth date by the 1970 draft lottery. It is difficult to see any systematic 
association between birth date and lottery number in the scatterplot. 
Clever graphics can emphasize the association, as in the figure. But a 
probability calculation is needed to learn whether the observed associa- 
tion is larger than can reasonably be attributed to chance alone. 

In a random assignment of draft numbers to birth dates, we would 
expect the correlation to be close to 0. The observed correlation for the 








400 r 






^ r ^ . 

. + 

. *7 

100 200 300 



FIGURE 10. Data from the 1970 draft lottery reveal a slight 
negative correlation, with birth dates near the end of the year most 
likely to have low draft numbers. The trend can be seen more 
readily by plotting the median draft numbers for each month. 
The plot of monthly medians connected by line segments to 
display the trend, called a median trace, is a common tool used 
to highlight patterns in scatterpiots of a response variable against 
an explanatory variable. 

1970 lottery was r = -0.226, showing that men born later in the year 
tended to get lower draft numbers. Common sense alone cannot decide 
if r = -0.226 means that the lottery was not random. After all, the 
correlation in a random lottery will almost never be exactly 0. Perhaps 
that r = -0.226 is within the range of values that could plausibly occur 
due to chance variation alone. 

To resolve this uncertainty we compare the observed -0.226 to a ref- 
erence distribution, the sampling distribution of r in a truly random 
lottery. We find that a truly random lottery would almost never pro- 
duce an r as far from zero as the r observed in 1970. The probability 
calculation tells us what common sense could not that r = -0.226 is 
a large effect, a surprising effect in a random lottery. This convinces us 
that the 1970 lottery was biased. Investigation disclosed that the cap- 
sules containing the birth dates had been filled a month at a time and 
not adequately mixed. Later dates remained near the top and tended 


to be drawn earlier. (Fienberg 4 gives more detail about the 1970 draft 
lottery, including extensive statistical analysis of the outcome.) 

Questions like "Is this a large outcome?" or "Is this a surprising re- 
sult?" come up often in analyzing data. It is quite natural to give an 
answer by comparing the individual outcome to a reference distribu- 
tion, as we informally compare the birth weight of a child to the dis- 
tribution of birth weights of all children. Students should certainly be 
encouraged to recognize the role of chance variation and to assess "sig- 
nificance" informally by comparing an individual outcome to a suitable 
reference distribution. If probability and computer simulation are be- 
ing developed, the comparison can be put in the language of probability 
and sampling distributions. But formal "tests of hypotheses" need not 
appear in the school curriculum. 

There are several reasons for this. The mechanics of stating hy- 
potheses, calculating a test statistic, and comparing with tabled values 
effectively conceal the reasoning of significance tests. The reasoning it- 
self is somewhat difficult and full of subtleties. Effective examples of 
the use of significance tests are more removed from everyday experience 
than opinion polls and similar examples of confidence statements. An 
understanding of data and chance, and the development of quantitative 
reasoning in general, is better served by concluding the study of statis- 
tics in the schools with probability, sampling distributions, confidence 
intervals, and a continuing emphasis on using these tools in reasoning 
about uncertain data. 


Statistics and probability are the sciences that deal with uncertainty, 
with variation in natural and man-made processes of every kind. As 
such they are more than simply a part of the mathematics curriculum, 
although they fit well in that setting. Probability is a field within math- 
ematics. Statistics, like physics or economics, is an independent disci- 
pline that makes heavy and essential use of mathematics. 

Statistics has some claim to being a fundamental method of inquiry, 
a general way of thinking that is more important than any of the spe- 
cific facts or techniques that make up the discipline. If the purpose 
of education is to develop broad intellectual skills, statistics merits an 
essential place in teaching and learning. Education should introduce 
students to literary and historical methods; to the political and social 
analysis of human societies; to the probing of nature by experimental 
science; and to the power of abstraction and deduction in mathematics. 
Reasoning from uncertain empirical data is a similarly powerful and 
pervasive intellectual method. 


This is not to say that detailed instruction in specific statistical meth- 
ods for their own sake should be prominent in the school curriculum. 
Indeed, they should not. But statistical thinking, broadly understood, 
should be part of the mental equipment of every educated person. We 
can summarize the core elements of statistical thinking as follows: 

1 . The omnipresence of variation in processes. Individuals are vari- 
able; repeated measurements on the same individual are variable. 
The domain of a strict determinism in nature and in human af- 
fairs is quite circumscribed. 

2. The need for data about processes. Statistics is steadfastly empir- 
ical rather than speculative. Looking at the data has first priority. 

3. The design of data production with variation in mind. Aware 
of sources of uncontrolled variation, we avoid self-selected sam- 
ples and insist on comparison in experimental studies. And we 
Introduce planned variation into data production by use of ran- 

4. The quantification of variation. Random variation is described 
mathematically by probability. 

5. The explanation of variation. Statistical analysis seeks the sys- 
tematic effects behind the random variability of individuals and 

Statistical thinking is not recondite or removed from everyday expe- 
rience. But it will not be developed in children if it is not present in 
the curriculum. Students who begin their education with spelling and 
multiplication expect the world to be deterministic; they learn quickly 
to expect one answer to be right and others wrong, at least when the 
answers take numerical form. Variation is unexpected and uncomfort- 
able. Listen to Arthur Nielsen 16 describing the experience of his market 
research firm with sophisticated marketing managers: 

. . . Too many business people assign equal validity to all numbers printed on 
paper. They accept numbers as representing Truth and find it difficult to work 
with the concept of probability. They do not see a number as a kind of shorthand 
for a range that describes our actual knowledge of the underlying condition. For 
example, the Nielsen Company supplies to manufacturers estimates of sales 
through retail stores. ... I once decided that we would draw all charts to show 
a probable range around the number reported; for example, sales are either up 
3 percent or down 3 percent or somewhere in between. This turned out to 
be one of my dumber ideas. Our clients just couldn't work with this type of 
uncertainty. They act as if the number reported is gospel. 

The ability to deal intelligently with variation and uncertainty is the 
goal of instruction about data and chance. There is some evidence 
that instruction actually improves this ability. Nisbett et al. 17 describe 


research on teaching various kinds of reasoning. They note that in- 
struction in probability and statistics increases the willingness to con- 
sider chance variation even when the instruction is of a traditional kind 
that makes no attempt to apply probabilistic reasoning in unstructured 
settings. Here is a typical example: 

[Subjects were asked] to explain why a traveling saleswoman is typically dis- 
appointed on repeat visits to a restaurant where she experienced a truly out- 
standing meal on her first visit. Subjects who had no background in statistics 
almost always answered this problem with exclusively nonstatistical, causal an- 
swers such as "maybe the chefs change a lot" or "her expectations were so high 
that the food couldn't live up to them." Subjects who had taken one statistics 
course gave answers that included statistical considerations, such as "very few 
restaurants have only excellent meals, odds are she was just lucky the first time," 
about 20 percent of the time. 

Nisbett and his colleagues find it striking that instruction of a quite ab- 
stract kind does have an effect on thinking about everyday occurrences. 
The effect is stronger when instruction points out the applicability of 
statistical ideas in everyday life, as school instruction should certainly 
do. This is evidence that we are in fact dealing with a fundamental 
and generally applicable intellectual skill. Nisbett also reports research 
showing that training in deterministic disciplines, even at the graduate 
level, does not similarly improve everyday statistical reasoning. This is 
evidence that we are dealing with an independent intellectual method. 

Why teach about data and chance? Statistics and probability are use- 
ful in practice. Data analysis in particular helps the learning of basic 
mathematics. But, most important, it is because statistical thinking is 
an independent and fundamental intellectual method that it deserves 
attention in the school curriculum. 


1. BBN Laboratories. ELASTIC and Reasoning Under Uncertainty. Final Report, 1989, 
p. 30. 

2. Barnett, Vic. Comparative Statistical Inference, Second Ed. New York, NY: John 
Wiley & Sons, 1982. 

3. Efron, Bradley. "Computers and the theory of statistics: Thinking the unthinkable." 
SIAM Review, 21 (1979), 419-437. 

4. Fienberg, Stephen E. "Randomization and social affairs: The 1970 draft lottery." 
Science, 171 (1971), 255-261. 

5. Garfield, Joan and Ahlgren, Andrew. "Difficulties in learning basic concepts in prob- 
ability and statistics: Implications for research." Journal for Research in Mathematics 
Education, 19 (1988), 44-63. 

6. Gastwirth, Joseph. "The statistical precision of medical screening procedures: Ap- 
plication to polygraph and AIDS antibodies test data." Statistical Science, 2 (1987), 


7. Gnanadesikan, Mrudulla; Schaeffer, Richard; Swift, James. The Art and Techniques 
of Simulation. Palo Alto, CA: Dale Seymour Publishers, 1986. 

8. Jones, L.V. (Ed.). The Collected Works of John W. Tukey. Vol. 3: Philosophy and 
Principles of Data Analysis, 1949-1964; Vol. 4: Philosophy and Principles of Data 
Analysis, 1965-1986. Monterey, C A: Wadsworth & Brooks/ Cole, 1986. 

9. Kruskal, William. "Miracles and statistics: The casual assumption of independence." 
Journal of the American Statistical Association, 83 (1988), 929-940. 

10. Landwehr, James and Watkins, Ann. Exploring Data. Palo Alto, CA: Dale Seymour 
Publishers, 1986. 

1 1 . Landwehr, James; Watkins, Ann; Swift, James. Exploring Surveys and Information 
from Samples. Palo Alto, CA: Dale Seymour Publishers, 1987. 

12. Mathematical Sciences Education Board. Reshaping School Mathematics: A Philos- 
ophy and Framework for Curriculum. National Research Council. Washington, DC: 
National Academy Press, 1990. 

13. Moore, David and McCabe, G. Introduction to the Practice of Statistics. New York, 
NY: W.H. Freeman, 1989. 

14. National Council of Teachers of Mathematics. Curriculum and Evaluation Standards 
for School Mathematics. Reston, VA: National Council of Teachers of Mathematics, 


1 5. Newman, Claire; Obremski, Thomas; Schaeffer, Richard. Exploring Probability. Palo 
Alto, CA: Dale Seymour Publishers, 1986. 

16. Nielsen, Arthur C., Jr. "Statistics in marketing." In Easton, G.; Roberts, Harry 
V.; Tiao, George C. (Eds.): Making Statistics More Effective in Schools of Business. 
Chicago, IL: University of Chicago Graduate School of Business, 1986. 

17. Nisbett, Richard E.; Fong, Geoffrey T.; Lehman, Darrin R.; Cheng, Patricia W. 
"Teaching reasoning." Science, 238 (1987), 625-631. 

18. Rubin, Andee; Bruce, Bertram; Rosebery, Ann; DuMouchel, William. "Getting an 
early start: Using interactive graphics to teach statistical concepts in high school/' 
Proceedings of the Statistical Education Section. American Statistical Association, 

19. Rubin, Andee and Rosebery, Ann. "Teachers' misunderstandings in statistical rea- 
soning: Evidence from a field test of innovative materials." In Hawkins, Ann (Ed.): 
Training Teachers to Teach Statistics. Proceedings of an International Statistics In- 
stitute Roundtable, July 1988. 

20. Tversky, Amos and Kahneman, Daniel. "Belief in the law of small numbers." Psy- 
chological Bulletin, 76 (1971), 105-110. 

21. Tversky, Amos and Kahneman, Daniel. "Extensional versus intuitive reasoning: The 
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We encounter patterns all the time, every day: in the spoken and 
written word, in musical forms and video images, in ornamental design 
and natural geometry, in traffic patterns, and in objects we build. Our 
ability to recognize, interpret, and create patterns is the key to dealing 
with the world around us. 

Shapes are patterns. Some shapes are visual, evident to everyone: 
houses, snowflakes, cloverleafs, knots, crystals, shadows, plants. Others, 
like eight-dimensional kaleidoscopes or four-dimensional manifolds, are 
highly abstract and accessible to very few. 

"The increasing popularity of puzzles and games based on the in- 
terplay of shapes and positions illustrates the attraction that geometric 
forms and their relations hold for many people/' observed geometer 
Branko Grlinbaum. "Patterns are evident in the simple repetition of a 
sound, a motion, or a geometric figure, as in the intricate assemblies of 
molecules into crystals, of cells into higher forms of life, or in count- 
less other examples of organizational hierarchies. Geometric patterns 
can serve as relatively simple models of many kinds of phenomena, and 
their study is possible and desirable at all levels." 

But despite their fundamental importance, students learn very little 
about shapes in school. The study of shape has historically been sub- 
sumed under geometry (literally "earth measurement"), which for a long 
time has been dominated by postulates, axioms, and theorems of Euclid. 


Just as Shakespeare is not sufficient for literature and Copernicus is 
not sufficient for astronomy, so Euclid is not sufficient for geometry. 
Like scholars in all times and places, Euclid wrote about the concepts 
of geometry that he knew and that he could treat with the methods 
available to him. Thus he did not write about the geometry of maps, 
networks, or flexible forms, all of which are of central importance today. 

Shape is a vital, growing, and fascinating theme in mathematics with 
deep ties to classical geometry but goes far beyond it in content, mean- 
ing, and method. Properly developed, the study of shape can form a 
central component of mathematics education, a component that draws 
on and contributes to not only mathematics but also the sciences and 
the arts. 

Like many other important concepts, "shape" is an undefinable term. 
We cannot say precisely what "shape" means, partly because new kinds 
of shapes are always being discovered. We assume we know what shapes 
are, more or less: we know one when we see one, whether we see it with 
our eyes or in our imaginations. 

But we know much more than this. We know that shapes may be alike 
in some ways and different in others. A football is not a basketball, but 
both are smooth closed surfaces; a triangle is not a square, but both are 
polygons. We know that shapes may have different properties: a triangle 
made of straws is rigid, but a square made of straws is not. We know 
that shapes can change and yet be in some way the same: our shadows 
are always our shadows, even though they change in size and contour 
throughout the day. 

In the study of shape, our goals are not so very different from those of 
the ancient Greek philosophers: to discover similarities and differences 
among objects, to analyze the components of form, and to recognize 
shapes in different representations. Classification, analysis, and repre- 
sentation are our three principal tools. Of course, these tools are closely 
interrelated, so distinctions among them are to some extent artificial. Is 
symmetry a tool for classifying patterns or a tool for analyzing them? 
In fact, it is both. Nevertheless, it is helpful to discuss each of these 
tools separately. 


One of the great achievements of ancient mathematics was the discov- 
ery that there are exactly five convex, three-dimensional shapes whose 
surfaces are composed of regular polygons, with the same number of 
polygons meeting at each corner. These shapes, known as the regular 
polyhedra, are shown in Figure 1. This discovery so excited the imag- 
ination of the ancients that Plato made these shapes the cornerstone 
of his theory of matter (see his dialogue Timaeus), and Euclid devoted 

SHAPE 141 

FIGURE 1. The five regular polyhedra. Each is composed of a single type of regular 
polygon, with the same number of polygons meeting at each comer. The tetrahedron, 
octahedron, and icosahedron are made of triangles, the cube is made of squares, and the 
dodecahedron is made of pentagons. 

much of Book XIII of his Elements to their construction. They have 
lost none of their fascination today. 

It is easy today to underestimate the significance of the discovery of 
the regular polyhedra. In its time it was a major feat of mathematical 
imagination. In the first place, in order to count the number of ob- 
jects of a certain kind you have to be aware that they are "of a certain 
kind." That is, you must recognize that these objects have properties 
that distinguish them from other objects and be able to characterize 
their distinguishing features in an unambiguous way. Second, you must 
be able to use these criteria to find out precisely which objects satisfy 
them. No one knows just how the ancients made their discovery, but it 
is easy for young children today, especially if they have regular polygons 
to play with, to convince themselves that the list of regular polyhedra is 
complete (Figure 2). 

The key ingredients of mathematical classification were already in use 
thousands of years ago: characterizing a class of objects and enumerat- 
ing the objects in that class. What has changed throughout the centuries, 
and will continue to change, are the kinds of characterizations that seem 
important to us and the methods that we use for enumeration. Figure 
3 shows several classes of objects that can be grouped together from 
a mathematical point of view. Examples such as these can stimulate 
student discussion: What properties characterize each class? Are there 
different ways to classify these objects? What other objects belong to 
these classes? We mention here a few of the classification schemes that 
have proved effective in many applications. 

Congruence and similarity. Two objects are congruent if they are ex- 
actly alike down to the last detail, except for their position in space. 
Cans of tomato soup (of the same brand) in a grocery store, square tiles 
on a floor, and hexagons in a quilt pattern are all familiar examples 
of congruent figures. Two objects are similar if they differ only in po- 
sition and scale. Similarity seems to be a very fundamental concept. 
Preschoolers understand that miniature animals, doll clothes, and play 
houses are all small versions of familiar things. The fact that even such 
voune children know what these tinv obiects are supposed to represent 




FIGURE 2. There are only five regular polyhedra because there are only five arrange- 
ments of congruent, regular polygons about a point that can be folded up to make a convex 
polygonal vertex. Here we see the five arrangements, together with their completion as 
patterns that can be folded up to make the entire polyhedron. 

shows that they intuitively understand change of scale. Building and 
taking apart scale models of towers, bridges, houses, shapes of any kind 
give the child of any age a firm grasp of this idea. 

Symmetry and self-similarity. A square is symmetrical: if you rotate 
it 90, 180, 270, or 360 about its center, it appears unchanged. Also, 
it has four lines of mirror symmetry across which you can reflect it onto 
itself (Figure 4). It is easy to think of other objects that have the same 
symmetries, or self-congruences, as the square: the Red Cross symbol, 
a bracelet with four equally spaced beads, a circle of four dancers, and 
a four-leaf clover (without its stem) are a few examples. Symmetry 
classifies objects according to the arrangement of their constituent parts. 



FIGURE 3. Examples of solid objects grouped into useful classes. 
What do the shapes in each class have in common? 

This can be rather subtle; for example, the two polyhedra in Figure 3b 
have the same symmetries. 

Just as congruence leads to symmetry (which is just another name 
for self-congruence), so similarity extends naturally to self-similarity. 
"The basic fact of aesthetic experience," according to art historian E.H. 
Gombrich, 9 "is that delight lies somewhere between boredom and con- 
fusion." Perhaps this is one of the reasons why fractals and other self- 
similar figures are generating so much excitement. 

"Beauty is truth, truth beauty," wrote the poet John Keats. Self- 
similarity has recently been recognized as a profound concept in nature. 
The awarding of a Nobel prize for the formulation of "renormalization 
groups" and the current worldwide cross-disciplinary interest in chaos 
theory indicate the profound implications of similarity and scale for 
science and mathematics. The study of scaling has stimulated (and been 
stimulated by) the study of fractals and other self-similar geometrical 

Combinatorial properties. Congruence and similarity are metric con- 
cepts: they can be altered by changing lengths or angles. But some other 

FIGURE 4. If a square is rotated 
90, 180, 270, or 360 about its 
center, it appears unchanged. Also, 
it has four lines of mirror symmetry 
across which you can reflect it onto 



FIGURE 5. In torus tic-tac-toe the opposite sides of the board are identified that is, 
considered to be the same. It is as if the board were rolled into a cylinder, which was 
then bent around to form an inner-tube shape that mathematicians call a torus. Can you 
tell which of these positions are equivalent in the torus-shaped game? 

properties of shapes remain intact under such transformations. For 
example, the numbers of edges and vertices of a polygon are not altered 
if we stretch or bend the polygon. Thus the three hexagons of Figure 7 
are all hexagons, even though they are neither congruent nor similar: a 
hexagon is any closed loop made of six line segments. Being a hexagon 
is a combinatorial property of a polygon. 

Roughly speaking, the combinatorial properties of a shape are the 
things we can count and the way they are fitted together. Thus from 
the combinatorial point of view, the shapes in Figure 3a are equiva- 
lent, since each has 6 faces, 8 vertices, and 1 2 edges connected to each 
other in the same way. Network problems often involve combinatorial 
problems. For example, if we want to design a linking system for the 
computers in a building, we are concerned first with finding the possible 
arrangements of links and nodes that can provide the connections we 
want, and only then need we consider how long the cables will have to 

Topology. Topological equivalence is even more general than combi- 
natorial equivalence. From the standpoint of topology, all polygons are 
loops and all convex polyhedra are alike. Piaget argued that topological 
concepts occur prior to metric ones in child development; a child may 
recognize a loop before distinguishing among kinds of loops, such as 
circles and triangles. Being a loop, as opposed to a knot, is a topological 
property of shape. 

Topology in school is often described as "rubber sheet geometry." It 
yields many excellent examples that can enlarge a child's concept of 
the flexibility of shape. In rubber sheet geometry, the shapes of Figure 
3c are indistinguishable because each can be deformed into the other. 
Knots, of the boy and girl scout variety, are an excellent subject for 
hands-on study. 12 Children can learn to play tic-tac-toe on a torus and 



other delightful games that require geometrical mental gymnastics (Fig- 
ure 5). 

With complexity of structure, topological classification necessarily be- 
comes more sophisticated. Here computer visualization can be a useful 
tool. Older students can appreciate the concept of orientation, which 
characterizes the difference between a cylinder and a Mobius band (ori- 
entable and non-orientable), and the concept of genus, which charac- 
terizes the topological difference between a sphere and a torus (genus 
zero and genus one). Understanding such concepts enriches greatly the 
study of science and design as well as mathematics. 


Shapes need names. One of the most fundamental uses of language 
is to assign names to things. Naming is a primitive concept that is 
echoed in our myths as well as in many contemporary religious practices. 
Naming is the first step toward knowing, whether it is the name of a 
person or the name of a shape. We cannot think about shapes (or 
anything else for that matter) or explain our ideas to others if we do 
not use names. Learning technical names is sometimes disparaged as a 
rote activity, but such objections miss the point. Technical names are 
usually not arbitrary; they encode the conceptual framework in which 
we organize the things we are naming. 

For example, in English-speaking countries, last names indicate the 
family and first names designate an individual in a family. Thus Mary 
Jones is a person named Mary who is a member of the Jones family. 
The names of shapes serve similar functions: a tetrahedron is a mem- 
ber of the polyhedron family, a representative of the subfamily of those 
polyhedra that have four faces (see Figure 6). When we use the word 

FIGURE 6. Mary Jones is a member of the Jones 
family, and the tetrahedron is a member of the 



"tetrahedron" to name a shape, we are at the same time locating it in 
its family tree and describing it in a meaningful way. 

Although classification requires precision, there is no single "right" 
way to classify shapes. Shapes are classified into families and subfam- 
ilies in many different ways, depending on the properties that interest 
us. For example, the discovery that the orbits of the planets around the 
sun are ellipses, and not circles, revolutionized the study of astronomy; 
from this standpoint circles and ellipses are completely different. But 
one of the great achievements of the ancients was the discovery that 
both circles and ellipses are conic sections and in that sense are the 

From the point of view of topology, the distinction between shapes 
that enclose regions, like balls, and shapes that have holes in them, 
like bagels, is fundamental; within these broad classes, all shapes are 
alike. But a football player would not be happy with a basketball as a 
substitute, nor would a basketball player be willing to make do with a 
baseball, because the individual kinds of balls have crucially different 
properties. As another example of cross-classification, architects know 
that it is important to build houses that are sturdy, not houses that might 
collapse. This concern transcends other ways that houses are commonly 
classified, such as large and small, single story or multistory, rectangular, 
or dome-like. 

Classification skills develop gradually. Very young children learn to 
recognize a great many shapes without being formally taught. Their 
world is literally made of shapes: shapes that hold things, such as bowls 
and bags and baskets; shapes to play with, such as balls and puzzles and 
blocks; shapes to use, such as chairs and spoons and beds. Thousands 
of shapes are part of children's lives. Later, in school, children learn 

FIGURE 7. Three hexagons that are 
important in chemistry. The planar 
hexagon (a) occurs in benzene (see 
also Figure 30 below). The hexagons 
in (b) and (c) are intended to be 
nonplanar; both are conformations of 
cyclohexane. Hexagons made out of 
flexible straws can easily assume any 
of these shapes. 



FIGURE 8. Four natural spirals: (a) leaves of the sago palm, (b) horns of a mountain 
sheep, (c) glycerin mixed with food coloring and ink, (d) the chambered nautilus. The 
common shape suggests a common creative mechanism, despite the striking differences 
in material, scale, and natural forces. 

names for some of them, such as circles, spheres, polygons, and some 
simple polyhedra. 

Alas, in our schools identification and classification of shapes usually 
stop just at the point where they can begin to be really interesting 
where they begin to explore structures in three-dimensional space. How 
many people realize that even polygons that are not flat can be interesting 
and important? Many molecules have polygonal shapes, but often these 
polygons are crumpled and their conformations are the key to their 
chemical properties (Figure 7). Besides finite polygons and polygons 
whose edges don't cross, there are zig-zag, star, and helical polygons. 
By broadening the definition of polygon to include any closed loop, we 
may also study knots. In addition to their obvious practical importance 
for tying things, knots enter into the design of networks such as clover- 
leafs and are helpful in understanding the structure of some biological 
molecules. Soap bubbles, soap films, and froths are also endless sources 
of fascinating geometrical principles. 

The study of polyhedra can be extended from simple shapes that are 
easy to construct to others, such as star polyhedra, that are more com- 
plex. Equally important are patterns, such as tilings of the plane, that 
are beautiful as well as useful. The helix and the spiral are fundamental 


to biology and astronomy as well as to mathematics. But even to- 
day, when "double helix" has become almost a household phrase, few 
people realize that there is a fundamental difference between a helix, 
which twists around an axis at a constant distance from it, and a spiral 
(Figure 8). Most so-called spiral staircases are really helical, for obvious 
practical reasons. Imagine what we would be like if our DNA wound 
itself in spirals, or what the universe would be like if galactic spirals 
were helices! 


In order to interpret and create patterns in today's image-packed 
world, it is not enough just to recognize similarities and differences; 
we also need to analyze them. This leads us to investigate the way that 
large shapes are built of smaller ones and to recognize patterns and their 

When children make shapes out of blocks or Legos, they often imitate 
the diverse compositions that they see around them (Figure 9). Nature 
too creates patterns. Like man-made patterns, natural patterns appear 
at many levels: atoms are organized into molecules, while molecules are 
organized into crystals and cells, which in turn are often the subunits 
of still higher-level organization. 

When we examine patterns carefully, we find that the same forms 
and arrangements appear over and over again, even when the objects 

FIGURE 9. Many shapes are built from smaller ones. The 
reinforcing beams in a bridge illustrate how repeated patterns are 
used in engineering and architecture, as in nature, to form a whole 
out of parts. 

SHAPE 149 

FIGURE 10. Young children can investigate the ways in which 
polygons can be fitted together to tile a plane surface. 

involved are very different. 16 This is not just a coincidence. The ge- 
ometry of most patterns is governed by a very few basic principles of 
formation, growth, and development. For example, in his fascinating 
book Patterns in Nature? Peter Stevens discusses several ways in which 
natural patterns are generated, such as stress, branching, meandering, 
partitioning, close packing, and cracking. The results of these modes 
of formation are remarkably similar, despite the variety of materials on 
which they operate (see Figure 8). 

Important aspects of pattern formation can be grasped by exploring 
the ways in which copies of objects can be packed together. Students 
quickly discover that there are only a very few ways to do this. This 
fundamental property of shape can be studied at many levels. For ex- 
ample, it can be studied intuitively and "hands-on" when the objects 
being packed are circles or easy-to-construct polygons such as triangles, 
quadrilaterals, and hexagons (Figure 10). Older children can experi- 
ment with less regular forms and discover some surprising things, such 
as the fact that any quadrilateral, even one that is not convex, will tile 
the plane (Figure 11). (This is a surprising but very simple consequence 
of the fact that the sum of the measures of the angles of a quadrilateral 
is 360.) High school students can study deeper properties of sphere 
packing and tilings, such as their symmetry and how they can be gener- 
ated. (Griinbaum and Shephard's Tilings and Patterns 1 is the definitive 
resource for material on tilings.) 


FIGURE 1 1 . Any quadrilateral will tile the plane, because the sum of the measures of 
its angles is 360, which is the same as the total number of degrees around each vertex. 
So four copies of a quadrilateral arranged around a point with each angle used once will 
fit just perfectly. 

Discovering Symmetry 

One of the most striking things about patterns of many kinds is their 
symmetry, and this symmetry is an important tool in their analysis. A 
pattern is something that repeats in some sense; symmetry is the concept 
that makes that sense precise. 

The study of symmetry begins by decomposing figures into congruent 
parts. Although some shapes do not at first appear to be made of smaller 
parts, it is often helpful to think of them as if they were. For example, 
mirror lines divide a square into eight congruent sectors, which the 
symmetries of the square permute. This decomposition helps us study 
the way symmetries work. In particular, it reveals that symmetry is self- 
congruence. It is this self-congruence that we consider beautiful and 
that makes symmetry a meaningful organizing principle in the analysis 
of structure. 

Young children learn quite easily to recognize symmetry, not only in 
squares and butterflies, but also in animals, flowers, household utensils, 
toys, buildings, and arrays of every kind. Symmetry can be found almost 
everywhere. Older children can get great pleasure, and gain great insight, 
by creating symmetrical patterns and discovering the rules that govern 

One of the most interesting but underappreciated techniques for ex- 
ploring patterns is paper folding. We are all familiar with the pretty 

SHAPE 151 

patterns that result when folded paper is cut and then unfolded. The 
snowflakes, chains of dolls, and other repeating patterns that appear 
are not created by magic but are simple consequences of the geome- 
try of reflection. Many geometrical constructions, and even aspects of 
number theory (some of them decidedly nontrivial), can be represented 
by unfolded designs. Conversely, many interesting three-dimensional 
shapes can be created by folding paper: the polyhedral nets of Figure 2 
are one example; origami puzzles are another. Paper-folding problems 
stimulate the geometrical imagination in many ways. 

Mirror Geometry 

Mirrors can be used to study the principles of reflection. In particular, 
building a kaleidoscope is an excellent way to discover how reflections 
interact to generate the orderly arrangements that we call kaleidoscopic 
patterns. The kaleidoscope is much more than a toy: it is a lesson 
in mirror geometry. Even one mirror has much to teach us: adults as 
well as children are challenged by the "mirror cards" used in elementary 
school classes. The kaleidoscope is more complex, but it too is based 
on the principles of reflection in a mirror. 

To explore the operation of a simple kaleidoscope, you just need two 
rectangular pocket mirrors and some tiny colored objects bits of plastic 
or glass will do very well. Tape the two mirrors together along one 
edge, with their reflecting surfaces facing each other. Place the objects 
on a table, between the standing mirrors (Figure 12). If you look in 
the mirrors you will see the objects repeated in a delightful pattern. 
A little experimentation will show that some angles produce lovelier 
configurations than others. Only certain angles produce, in the words 
of the kaleidoscope's inventor, Sir David Brewster, "a perfect whole" 
a finite number of identical regions arranged in a circular pattern. By 
playing with the mirrors, it is not difficult for children to discover which 
angles produce this perfect kaleidoscopic image. By doing so they will 
have learned an important lesson in the modern study of shape. 

Reflections generate patterns with a finite number of subunits, pat- 
terns that have rotational as well as mirror symmetry. The rotations 
and reflections can be performed one after the other, always leaving the 
"perfect whole" apparently unchanged. Formally, such a system of mo- 
tions is known as a symmetry group. Many properties of shapes can be 
analyzed by studying their symmetry groups; indeed, for more than a 
century this strategy has been a guiding principle in the study of geome- 
try. By using a kaleidoscope, students can understand this fundamental 
idea by direct experience without making a lengthy detour through the 
formal and abstract algebraic language in which it is usually expressed. 



FIGURE 12. The principle of the kaleidoscope is discovered by playing with two hinged 
pocket mirrors. The objects appear repeated in infinitely varied patterns, but as the angle 
between the mirrors is changed, some patterns reveal greater symmetry (and beauty) than 

FIGURE 13. A pyrite crystal. The lines on the cube's faces 
indicate that the crystal's internal structure lacks some of the 
symmetries of the cube. 



FIGURE 14. A cubic kaleidoscope 
can be made by placing mirrors or 
reflecting mylar on the inside of three 
sides of one of the tetrahedral sectors 
into which the cube is divided by its 
mirror planes (a). The net for these 
three walls is shown in (b); it consists 
of half a square and a rectangle whose 
base is the length of the square's edge and whose height is the length of the square's 
diagonal. Cut along the dotted lines, and then tape the edges a and ft together (c). 
With the cut end down and parallel to a table, look at a piece of newspaper or other 
decorated material through the tetrahedron. You will see a decorated cube! By moving 
the tetrahedron along the plane surface, you will see a changing pattern on the cube. 

The symmetry of three-dimensional figures appears to be more intri- 
cate, but actually the principles are the same as in the two-dimensional 
case. For example, the symmetry of the cube includes reflections in two 
kinds of mirror planes and rotations about three kinds of axes. Younger 
students can learn a great deal about the symmetry of the cube by trying 
to decorate it in ways consistent with its symmetry. Older students can 
be challenged by the task of changing this symmetry by decoration. 

Such decorations appear in nature, where they provide clues to the 
structure of hidden patterns. For example, the pyrite crystal in Figure 
1 3 appears at first glance to be an ordinary cube, but closer inspection 
reveals striations on the cube's faces. These striations are consistent 
with some, but not all, of the symmetries of the cube. The reason 
for the striations, it turns out, is that the arrangement of atoms inside 
the crystal is less symmetrical than its external cubic form suggests. 
Consequently, the pyrite crystal is a cube with texture, or a decorated 

One of the more exciting and instructive exercises for older students is 
to make a cubic kaleidoscope. The cube is divided by its mirror planes 
into 48 congruent tetrahedra. If a model of one of these tetrahedra 
is lined with mirrors or some reflecting paper such as mylar, with the 
triangle belonging to the cube space removed and the opposite vertex 
snipped off, an entire cube is generated by the reflections. Reflecting 
mylar pasted onto cardboard or heavy paper will work well; only three of 
the four tetrahedral walls should be constructed so that you will be able 
to see inside. Figure 14 shows how to construct such a kaleidoscope. 18 

Using Symmetry 

If all we learn about symmetry is to identify it, we miss the whole 
point. Symmetry is an effect, not a cause. 19 Why are so many natural 
structures symmetrical? For example, what atomic forces ensure that 



FIGURE 15. Semiregular polyhedra are 
formed by using several kinds of regular 
polygons as faces, with the same arrange- 
ment at each vertex and all vertices inter- 
changeable by symmetry operations. 

the arrangements in crystals will be orderly? Although these are pro- 
found and largely unsolved problems, a good working answer was given 
over thirty years ago by James Watson and Francis Crick in describing 
their discovery of the structure of DNA: 22 

Wherever, on the molecular level, a structure of a definite size and shape has 
to be built up from smaller units ...the packing arrangements are likely to 
be repeated again and again and hence sub-units are likely to be related by 
symmetry elements. 

In other words, nature builds modular structures that organize them- 
selves according to certain rules. Repetition of the rules tends to lead 
to arrangements of modules that we call symmetrical. 

Polyhedra provide a wealth of excellent examples of arrangements 
that are repeated again and again. When you build a cube with card- 
board squares by attaching three squares to each corner, you are con- 
structing a shape that satisfies a certain packing arrangement: it must be 
made of congruent regular polygons, and it must have the same number 

SHAPE 155 

FIGURE 16. Convex deltahedra are 
formed from equilateral triangles ar- 
ranged with differing types of vertex 
arrangements: three, four, or five tri- 
angles may be joined at a vertex. 

at each corner. By generalizing this construction to other polygons, we 
obtain the five regular polyhedra (Figure 1). The arrangements can be 
further generalized to include the semiregular polyhedra (Figure 15), in 
which more than one kind of regular polygon can be used, and the con- 
vex deltahedra (Figure 16), all of whose faces are equilateral triangles 
but whose vertex arrangements need not all be the same. 17 

The cover design for the biological journal Virology contains an icosa- 
hedron. The story of the discovery of icosahedral symmetry in viruses 
and the ongoing efforts of scientists to link that symmetry to their sub- 
unit structures is very instructive. 17 Viruses are tiny capsules that con- 
tain an infective agent. The capsule is composed of protein subunits 
that group together to form a closed shell. Watson and Crick realized, 
in the course of early X-ray investigations into virus structure, that the 
shells of many viruses had polyhedral or helical forms. Subsequent stud- 
ies showed that the polyhedra were often icosahedra, and this suggested 
many attractive models for the arrangement of the protein subunits. But 
more recently these models have been found to be incorrect. The con- 
nection between packing arrangements and overall symmetry in viruses 
remains an unsolved problem. Problems such as these lead also to new 
developments in mathematics: they force mathematicians to rethink 
their definitions and to broaden the scope of their investigations. 


From earliest times the beautiful shapes that we call crystals have 
been a source of wonder and admiration. Why do they have polyhe- 
dral forms when most other natural structures do not? Quartz crystals 
were the first to be studied; at first they were thought to be pieces of 
permanently frozen ice. (It is instructive that our word "crystal" comes 
from the Greek word /c/?zara/Uog, which means ice.) By the seven- 
teenth century, scientists began to suspect that the shapes of crystals 
reflected an orderly, patterned, internal structure. Long before the de- 
velopment of modern atomic theory it was suggested that crystals are 
made of stacks of tiny spheres that represented the basic particles of the 
structure, whatever those might be. Later the particles were represented 
as tiny bricks (Figure 17). Sphere packings and bricks (not necessarily 
rectangular) are still important models for crystal structure. 



FIGURE 17. An 1822 concept of crystal structure in which various crystal shapes are 
imagined as being built from tiny rectangular bricks. 

Whether we use spheres or bricks, the important idea is that of an or- 
derly array. Let us explore this a little further. A one-dimensional lattice 
is a set of points equally spaced along a line. (Although we can draw only 
part of the set, we assume that it goes on forever.) All one-dimensional 
lattices are essentially alike, differing only in the spacing between points. 
But there are two basic kinds of two-dimensional lattices: one in which 
the points of the rows lie directly above one another, the other in which 
they are shifted horizontally (see Figure 18). Each point of a lattice 
"occupies" a certain portion of the plane, the region nearer to it than to 
any of the other lattice points. These regions, called Dirichlet domains, 
display the symmetry of the lattice in a corresponding brick model. The 
Dirichlet domains in two dimensions the bricks are always quadri- 
laterals or hexagons; within each lattice the regions about each of the 
points are congruent. 

Lattices describe the underlying symmetries of patterns. Draw a one- 
dimensional lattice on two or three different sheets of tracing paper, and 

SHAPE 157 

f* J 

FIGURE 18. The symmetry of 
two-dimensional lattices is dis- 
played by their Dirichlet domains, 
polygons centered at each lattice 
point which enclose the region of 
the plane closer to the enclosed lat- 
tice point than to any other. These 
polygons may be quadrilateral or 
hexagonal; for a given lattice they 
are all congruent. 

use them to create two-dimensional lattices. You will quickly discover 
that you can change the symmetry of the lattice by shifting the relative 
positions of the rows: you can check the symmetry by recalculating the 
Dirichlet domains. No matter what you do, the symmetry will always be 
of one of the five types shown in Figure 18. It is an important fact that 
every two-dimensional repeating pattern, whether it is an arrangement 
of points or ellipses or polygons, a wallpaper pattern, or an Escher-like 
tiling of the plane, can be interpreted as a decoration of the Dirich- 
let domains associated with a lattice that belongs to one of these five 
symmetry types. 

This simple observation raises a wealth of interesting questions. What 
kind of packing arrangements can we create if we replace the points by 
other shapes? What shapes can be fitted together without gaps to form 
orderly patterns? What do we mean by orderly? What are the possi- 
ble ways to extend arrays to three dimensions? It turns out that there 
are only a small number of solutions to problems such as these, which 
explains why the same patterns reappear so often in crystal structures, 
trusses, biological tissues, honeycombs, wallpaper, textiles, and tiled 

Three-dimensional lattices have been used by mathematicians and 
scientists, beginning in the nineteenth century, to try to explain the ar- 
rangements of atoms in crystals. In three dimensions there are 14 sym- 
metry types of lattices and 5 combinatorial types of Dirichlet domains 
(see Figure 19). 

It is difficult to overestimate the importance of play with cubes and 
other blocks. Even one year olds enjoy building taller and taller tow- 
ers and watching them fall down. Later, children use blocks to build 



FIGURE 19. There are five combinatorial types of Dirichlet domains for three- 
dimensional lattices. These five shapes are much less well known than the Platonic solids 
but are at least as important! 

houses, courtyards, and other structures. Young children are likely to 
have trouble building octahedra, but they can use small cubes to build 
larger cubes. The smallest composite cube is made up of 8 smaller ones; 
the next larger is made up of 27; by guessing how this series continues 
the child gains some understanding of volume. Older children of any 
age can also learn a lot from playing with cubes. 

Cubes are the prototypical three-dimensional tile, and many struc- 
tures, both mathematical and real, are based on it. It is worthwhile to 
try to build polyhedra out of cubes. For example, try building a regular 
octahedron by sticking sugar cubes together with glue. The larger you 
make your sugar octahedron (if it isn't too messy), the closer the stepped 
faces approximate smooth ones. Building polyhedra from cubes is thus 
a sophisticated lesson in volume measurement. It is instructive that 
H.S.M. Coxeter, in his classic work Regular Poly topes, 5 refers to the 
cube of any dimension as the "measure polytope." (The word "poly- 
tope" refers to the higher-dimensional analogues of polygons and poly- 


An important problem in many fields is how to divide a region into 
compartments of various shapes. An architect or designer partitions 
the interior of a building into rooms to serve certain purposes. We all 
fret over the most efficient way to pack a suitcase or the trunk of a 
car. A complex living object, such as a plant or a human being, has 
grown from a single cell that, in the early stages of growth, divided 
into "daughter" cells that grew and divided again. The study of how 
dividing cells organize themselves into tissues and then into organs is 



one of the most exciting frontiers of biology. Some of the issues relate 
to the geometry of dissection, compartmentalization, and subdivision. 

There are many interesting mathematical problems dealing with dis- 
section. One of the most famous theorems in this field says that any 
polygon can be divided into a finite number of pieces and reassembled 
to form a congruent copy of any other polygon of the same area. El- 
ementary school children enjoy the challenge of creating shapes with 
tangrams or other polygonal tiles; imagine the many challenging prob- 
lems and puzzles that could be devised for older children related to 
this dissection theorem. More advanced students can discover that the 
analogous theorem for polyhedra is false; this is another fascinating and 
important result. 

Another intriguing dissection problem is the creation of "rep-tiles," 
tiles that can be fitted together to form replicas of themselves (Figure 
20). Alternatively, we can create such tiles by subdividing one into 
smaller congruent copies of itself. To create a tiling by rep-tiles, think 
of the daughter tiles growing to the size of the original one and then 
subdividing again. Repeating this process over and over again, we cre- 
ate a tiling that is self-similar in a certain sense; many of these tilings 
have no lattice structure. Is the tiling of Figure 20 lattice or nonlattice? 

FIGURE 20. "Rep-tiles" are tiles that can be fitted together to 
form replicas of themselves. They build tilings that are self- 
similar and that, like this one, may have no lattice structure. Such 
tilings are of great interest today because they share many strange 
properties with some newly discovered crystalline materials. 



(This is not easy to answer!) Tilings without lattices are of great inter- 
est today among mathematicians and solid-state scientists because they 
share many strange properties with some recently discovered crystalline 
materials called quasicrystals. 

Combinatorial Tools 

Combinatorial properties of patterns are also very important because 
they provide clues to what is possible and what is not. For example, 
suppose we want to build a tetrahedron that is, a polyhedron with four 
faces. How should we start? Before cutting out polygons and trying to 
tape them together, let's reason out the possibilities. In the first place, all 
the polygons will have to be triangles, because as we build we will start 
with one polygon and attach another to each edge. If our first polygon 
had more than three edges, we would run out of polygons, since we only 
have four. So to build a tetrahedron we attach a triangle to each edge 
of our first triangle (Figure 2a), and then to make a closed polyhedron 
we must fold up the configuration so that the other triangles meet in 
a point. This means that the edges of the polyhedron must form a 
network of four triangles. 

We can go on from here to discuss properties (e.g., congruence) these 
triangles might have, but it is important to note that we have already 
made an important discovery: every tetrahedron is a combinatorial net- 
work of four triangles. Similar reasoning shows that there are two com- 
binatorial types of pentahedra, polyhedra with five faces (you can find 
them in Figure 6). There are exactly seven types of hexahedra (Figure 
21), including, of course, the cube. It is a challenge for students to 
discover why there are no more. 

FIGURE 2 1 . There are seven combinatorial types of hexahedra. 
Try to visualize them as three-dimensional polyhedra! 

SHAPE 161 

The combinatorial properties of shapes are sometimes more funda- 
mental than their metric properties. If we try to build a convex poly- 
hedron out of hexagons, we will never succeed: such polyhedra are 
combinatorially impossible. It's better to know this in advance! A few 
years ago a "World Sports Day" poster featured a giant soccer ball that 
appeared to be made entirely of hexagons. The designer did not realize 
that she had drawn an impossible figure! 

The fundamental theorem of combinatorial theory for polyhedra is 
Euler's Theorem, which is valid for every convex polyhedron (and some 
others): the sum of the number of faces plus the number of vertices is 
equal to the number of edges plus two. This can be written succinctly 

where F is the number of faces (or cells) of the network, V is the 
number of vertices, and E is the number of edges. (It is easy to verify 
this equation with the networks in Figures 1, 6, 19, and 27). Euler's 
Theorem is easy to discover (with guidance), easy to teach, and, for 
more advanced students, not difficult to use. The theorem and its many 
corollaries and generalizations are important tools for enumerating the 
combinatorial properties of objects. 


A third important tool in the study of shape is representation. In ev- 
eryday life as well as in science, mathematics, and art, we deal not only 
with shapes themselves but also with many kinds of representations of 
shapes models, photographs, drawings. The tools of representation 
include the ability to understand scale models; to read maps; to under- 
stand shadows, sections, and projections; to reconstruct shapes from 
their images; to draw accurately; and to use computer graphics. The 
underlying issue is the same in each case to determine the relation 
between a shape and its image or between different images of the same 


The simplest representation of a shape is a model of it, built to an 
appropriate scale. A spherical globe is a model of the earth, or of the 
moon, or of any planet. A globe is not an exact replica of the earth, 
but an approximate one that displays certain features of the earth quite 
well. It is approximate because it is perfectly round, which the earth 
is not. Besides, it is constructed on such a small scale that even our 
largest cities appear as tiny dots. But every child growing up in this 



Cylindrical with one 
standard parallel 

Conic with one 
standard parallel 

Azimuths 3 in contact at 
one point 

Cylindrical with two 
standard parallels 

Conic with two 
standard parallels 

Azimuths) intersecting globe 




FIGURE 22. Mapmakers use many different methods of projecting the globe to create 
flat maps. The choice of projection determines many of the map's features. 

SHAPE 163 

space age understands that the globe is a model of the earth. Most 
models are approximate in the sense that they ignore some details in 
order to present key features more vividly. Making a model entails 
making a selection of which features are to be emphasized; this point 
merits classroom discussion. 


Interesting questions about the relation between shape and image 
arise, for example, in the study of maps. Why do we use both globes and 
flat maps? The answer is simple: they are useful for different purposes. 
Although a globe and a flat map represent the same thing, namely the 
earth, they display its properties very differently. 

Flat maps can represent small regions of the earth quite well, since 
part of the surface of a spherical object can be closely approximated 
by a plane. But the representation gets worse and worse as you try to 
increase the area represented by the map. The relation between globes 
and flat maps leads quickly to very fundamental geometrical questions. 
You can't make a sphere by folding up a sheet of paper, so to make a 
flat map, you have to project the globe in some way. Mapmakers use 
several different projection methods (Figure 22). Thus a map of the 
earth is an approximation of a spherical surface, an approximation that 
gets worse and worse as the portion of the globe being mapped is in- 
creased. Every flat map necessarily distorts angles, areas, or both. Like 
a mathematician considering more general kinds of maps, every map- 
maker must compromise by deciding which features of representation 
are most important for particular purposes. 

Shadows and Lenses 

Shadows, perhaps the most familiar examples of images, are never- 
theless rather subtle because they distort contour as well as size. The 
interesting question is to determine what sorts of distortions can occur, 
and why. 

Young children can learn a great deal by observing their own shadows. 
To create a shadow, you need a light source (if you are outdoors, it is the 
sun), an object (you), and a screen (the ground or a wall). The shadow 
is your projection onto the screen, and what that projection looks like 
depends on the positions of the light, the object, and the screen. Older 
students can experiment, varying the positions of the light, the screen, 
and the object that blocks the light to produce the shadow. From this 
they can discover which properties of shapes are preserved and which 



FIGURE 23. The circle, the ellipse, 
and the parabola as shadows cast by a 
circle on screens in different positions. 

are lost under this kind of projection. For example, all the conic sections 
can be created as projected shadows of a circle (Figure 23). 

At a more advanced level we can think of shadows as maps in which 
only the outline of the map is retained. From this perspective the prin- 
cipal difference between a map of the earth and your shadow on the 
wall is the object being mapped. 

Lenses, too, distort shape but in more predictable ways. Eyeglasses, 
slide projectors, telescopes, microscopes, and cameras are only a few 
of the tools through which lenses enter our lives. Indeed, lenses in 
our eyes provide our only access to visual images. The study of lenses 
involves many principles of geometry that can be taught at every level 
from kindergarten through high school and beyond. 


In every culture and in every era artists have grappled with the prob- 
lem of representing three-dimensional shapes on two-dimensional sur- 
faces. The solutions they have found are, in many cases, the same as 
those of the mapmaker. For example, in Figure 24 the artist is literally 
making a map of the shape he sees before him. The device he is using 
is easy to make and can be used in the classroom with good results. 
Perspective drawing is another example of mapping. 

Before the camera became available to everyone, drawing was widely 
taught. Today very few people know how to draw accurately, and, con- 
sequently, they no longer notice things as carefully as they once did. 
A few years ago there was great embarrassment (or should have been) 
when Branko Griinbaum discovered that the icosahedral logo of the 


FIGURE 24. Albrecht Durer's sketch of an artist making a map of the shape he sees 
before him. 

Mathematical Association of America, which appeared on all of its pub- 
lications, was inaccurately drawn (Figure 25); this error had escaped de- 
tection for several years even though it was seen regularly by thousands 
of mathematicians. If visual illiteracy is so widespread even among 
professional mathematicians, future generations run the risk of really 
believing that they live in an Escher-like impossible world (Figure 26)! 
In his article on the misdrawn icosahedron Griinbaum 11 presented 
a small sample of his collection of badly drawn textbook figures (see 
Figure 27). Looking at them with a trained eye is enough to make 
one laugh or cringe (or both). But how many of us could do better? 
Indeed, how many teachers of mathematics can even draw a respectable 
cube? These gaffes presumably would not have occurred were authors 
and graphic artists more familiar with the principles and practice of 
perspective and projection. For many years technical drawing has been 
relegated to courses in the fine and industrial arts when, in fact, they 
are essential for all students. 

FIGURE 25. The icosahedral logos of the Mathematical Association of America: the 
old one is badly drawn; the new one is accurate. The error escaped detection for many 
years. Can you tell which is which? (Hint: In drawing a projection of a three-dimensional 
figure on a plane, parallel lines should stay parallel or else intersect in a single point.) 



FIGURE 26. Belvedere, by M.C. Escher, provides a visual com- 
mentary on the subtleties of representing a three-dimensional scene 
on a two-dimensional piece of paper. 

Image Reconstruction 

If the artist's problem is to represent a three-dimensional shape on a 
flat surface, the viewer's problem is to recognize what shape the image 
is supposed to represent. A visit to an art gallery is an exercise in image 
reconstruction. So is the physician's task of reading an X-ray or a space 

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FIGURE 27. Some badly drawn figures, all taken from published books dealing with 
geometry and related subjects. 

The painting, the X-ray, and the photograph are maps of shapes, 
which we need to be able to read "in reverse." This subject, closely 
related to the problem of visualization, is of great importance in the 
study of shape, but it has not been organized in a way that can be used 
in school. Here is a challenge: to bring to students of mathematics 
the wealth of material related to shadows, cross sections, and projec- 
tions. In addition to ideas now taught in art classes and in the industrial 
arts, students could learn about criteria for deciding whether a projec- 
tion is properly drawn (i.e., whether a diagram is in fact a projection 
of a three-dimensional form). They could learn the principles of the 
stereoscope and why stereoscopic pairs appear three-dimensional to us. 
They could also learn to deduce symmetry and topological properties of 
a three-dimensional shape from its two-dimensional representation. A 
discussion of optical illusions and "impossible" figures can lead to many 
important insights. Those familiar with the combinatorial properties of 
polyhedra can study their representation through planar graphs and can 
try their skill at reconstructing corresponding three-dimensional forms. 

Computer Graphics 

The computer is not a substitute for real three-dimensional mod- 
els. Images on the computer screen, even the so-called 3-D images, are 
meaningful only if the viewer has extensive prior experience with three- 
dimensional structures. On the other hand, computer graphics can be 
fascinating to students and can generate strong interest in the study of 
shape. Good software can thus be invaluable in the study of shape and 
should be used when appropriate. Moreover, every person should know 
something of the geometry that underlies computer graphics above all, 
coordinate geometry in order to use graphics packages intelligently and 

In summary, the creation of images and the reconstruction of shapes 


facets of representation can be organized under the concept of map- 
pings. "Real" maps are only one example; shadows, sections, images 
seen through lenses, images produced by projection, images produced 
by reflection, and images rendered by the graphic or photographic artist 
are others. As increasingly detailed images of the very large, the very 
small, and the formerly hidden are made visible by modern technology, 
the need to understand this broadened concept of mapping becomes 
increasingly urgent. 

Mapping is a major theme of contemporary mathematics because it 
provides a useful and illuminating way to organize relations among 
shapes and patterns (including very abstract ones). It also helps us to 
make our classification systems precise. Congruence and similarity can 
be described in the language of maps. For example, the shapes in Fig- 
ure 3a can be transformed into one another by a mapping that preserves 
their combinatorial structure; the shapes in Figure 3b are related because 
they have the same set of symmetries (which are mappings of the ob- 
jects on to themselves); the shapes of Figure 3c can be transformed into 
one another by a mapping that is a continuous deformation. 


Visualization is a broad subject with implications for many aspects 
of our lives. It is centrally important to all of mathematics and has 
been so throughout history. Mathematics made a great advance with 
the invention of numerals, which are visual representations of numbers. 
Certainly one of the major mathematical achievements of the last several 
hundred years was the development of analytic geometry, which enabled 
us to combine visual and formal mathematical thought. 

Obviously, visualization is very important in the study of shape. But 
it is also important for all of mathematics. To study change, we need to 
see it; to study data, we examine various graphical representations. We 
try to grasp the concept of higher dimension by drawing pictures and 
by making models. Even the properties of numbers can be illuminated 
by visual representation that is what the number line is for. But it 
is not true that we instinctively know how to "see" any more than we 
instinctively know how to swim. Visualization is a tool that must be 
cultivated for careful and intelligent use. 

It may be helpful to retell a very old story about Galileo's discovery of 
mountains and craters on the moon, a discovery that helped to change 
forever the way we view the universe and our place in it. "Following 
Aristotle, Europeans of the Middle Ages and the Renaissance believed 
that the moon was a perfect sphere, the prototypical shape not only of 
the visible planets and stars but of the entire universe," explains the art 
historian Samuel Edgerton. 6 



The problem, thus, was not to determine its shape, which all accepted, but to 
explain the mottled appearance of its surface, that "strange spottednesse," as 
Harriot called it. Some ancient authorities had explained the spots by arguing 
that the lunar surface was like a gigantic mirror reflecting the lands and seas 
of the earth. Others had claimed that the moon was composed of transparent 
substance with some internal denser matters giving off varying amounts of light. 

Galileo found another explanation: 8 

I have been led to the opinion and conviction that the surface of the moon is 
not smooth, uniform, and precisely spherical as a great number of philosophers 
believe it (and the other heavenly bodies) to be, but is uneven, rough, and full 
of cavities and prominences, being not unlike the face of the earth, relieved by 
chains of mountains and deep valleys. 

Thomas Harriot was an English astronomer who had also been looking 
through a telescope at the moon at the same time that Galileo made 
his discoveries. Harriot's sketches show, however, that the "strange 
spottednesse" did not look like mountains and valleys to him (Figure 

How could it happen that Harriot and Galileo, looking at the same ob- 
ject through comparable telescopes, did not "see" the same thing? True, 
Galileo was the greater genius, but this fact alone is not very illuminat- 
ing. Edgerton suggests a more persuasive reason: Galileo was a trained 
artist, skilled in the use of perspective and chiaroscuro, the rendering of 
light and shadow. Thus "Galileo did indeed have the right theoretical 
framework for solving the riddle of the moon's 'strange spottednesse.' 
Unlike Harriot, he brought to his telescope a special 'beholder's share' 
(as E.H. Gombrich would say); that is, an eyesight educated to 'see' the 
unsmooth sphere of the moon illuminated by the sun's raking light." 

FIGURE 28. Harriot's and Galileo's sketches of the lunar surface. 


FIGURE 29. The benzene ring was made visible on the atomic scale for the first time 
in 1988. This image was produced by a scanning tunnelling microscope. An educated 
eye can see the triangular patterns of electrons connecting pairs of the six carbon atoms 
in each ring. 

"Galileo's telescopic discoveries opened the eyes of Europeans every- 
where," continues Edgerton. And, as his notebooks show, "even Harriot 
'saw' shaded craters once he was aware of the Florentine's observations." 

Today, we all see mountains and valleys when we look at the moon. 
But would we see them if we didn't already know what we were supposed 
to see? And what do we "see" when we look at the images presented to 
us by modern technology? The educated "beholder's share" is just as 
essential today as it was in Galileo's time. "Whether the object is a virus 
seen through an electron microscope, a distant galaxy explored by radio 
telescope, or a fetus observed in the womb by means of ultrasound, 
theoretical assumptions have to be made before the raw data can be 
translated into an image," writes Hans Christian von Baeyer in a recent 
issue of The Sciences. 2 

Von Baeyer goes on to point out that this translation must be done by 
the educated eye as well as by the internal workings of the computer. A 
case in point is the first atomic-scale image of the "hexagonal" benzene 
rings, which was produced for the first time in 1988 (Figure 29). Can 
you see the hexagons? Or do you see some lumpy donuts? Or spher- 
ical triangles? Scientists were able to find the triangular traces of the 
hexagonal structure because they already knew that they were there. 

Rudolf Arnheim stressed the importance of visualization in science 
in his aptly titled Visual Thinking-. 1 

SHAPE 171 

The lack of visual training in the sciences and technology on the one hand and 
the artist's neglect of, or even contempt for, the beautiful and vital task of mak- 
ing the world of facts visible to the enquiring mind, strikes me, by the way, as 
a much more serious ailment of our civilization than the "cultural divide" to 
which C.P. Snow drew so much public attention some time ago. He complained 
that scientists do not read good literature and writers know nothing about sci- 
ence. Perhaps this is so, but the complaint is superficial Snow's suggestion 

that "the clashing point" of science and art "ought to produce creative chances" 
seems to ignore the fundamental kinship of the two. 

Like the weather, everyone talks about visualization, but no one does 
much about it. Visualization is not a simple matter: it is a deep subject, 
properly the domain of physiology and psychology and still not well 
understood. Nonetheless, it is easy to teach shape as an important first 
step in developing powers of visualization. The simplest way to teach 
students to visualize is to provide them with a rich background of hands- 
on experience with shapes of many kinds. A serious study of image 
reconstruction would also be a step in the right direction. 


Students should learn to recognize the patterns of shape, to under- 
stand the principles that govern their construction, and to be able to 
move easily back and forth between shapes and their images. Although 
the study of shape seems to fall between the cracks of traditional sub- 
jects, the new Curriculum and Evaluation Standards for School Mathe- 
matics 15 of the National Council of Teachers of Mathematics reflect an 
emerging consensus that this situation must be improved. 

The study of shape must be more than the sum of its parts; an inte- 
grated view of shape can help accentuate the whole subject. One possible 
approach is illustrated by the chart in Figure 30. 

Forging Connections 

Rethinking the subject as a whole provides us with an opportunity to 
forge substantive connections between the study of shape and the role 
of shape in the real world. We can take seriously Arnheim's plea for 
integrating art and science. We can also reduce the mystery of some 
of our contemporary technology. The principles of the electron micro- 
scope, the radio telescope, and ultrasound are not wholly beyond the 
scope of the K-12 curriculum; high school students can, if we wish, 
learn the foundation necessary to understand the action of these and 
other modern imaging techniques. 

Indeed a focus on shape makes many aspects of modern technology 
much more accessible than is commonly supposed. Here are just three 



A Structure for Shapes 
: Identification and Classification 



Plane Polygons 


Congruence; similarity 
Soap bubbles 


Zig-zag and star polygons; 



Tiling the plane with poly- 

Soap bubble clusters 


Helices, spirals, cylinders, 
tori, Mobius bands 
Escher-like tilings 

Simple crystal structures 
Orientation, genus texture 

: Analysis 


Mirror symmetry; rotational 



Paper-folding; patterns 

Constructing and decon- 
structing polyhedra 
Linear/volume measurement 

Making quilts and mosaics 

Two-mirror kaleidoscopes 

Symmetry of finite figures 

Dissection; puzzles 
Rep-tiles; fractals 
Natural patterns 

Regular and semiregular 


Angle measurement 

Tiling the plane with poly- 


Polyhedral kaleidoscopes 

Symmetry as an organizing 
principle; transformation 

Exploring fractals 

Scale in biology 

Euler's formula for polyhedra 

Fundamentals of plane and 

3-D geometry 

Lattices; elementary tiling 




Drawing, reading, and using 
simple maps 

Scale projectors 

Turtle geometry 

Representation and Visualization 



Relief maps and level curves 

The globe 
Shadow geometry 
Perspective drawing 

Telescope and microscope 
Plane coordinates 

Exploring geometry with the 


Cross-sections of 3-D shapes 

Geometry of the sphere; 
projections; maps 
Images and image reconstruc- 
tion; impossible figures 
Technical drawing; stereo- 

Lens geometry; the camera 
3-D coordinates 
More computer graphics 

FIGURE 30. An arrangement of topics related to shape that 
provides structure and coherence to what might otherwise appear 
as an arbitrary collection of quite disparate topics. 

SHAPE 173 

FIGURE 31. The structure of crystalline 
silicon. It is made entirely of zig-zag hexa- 
gons. This is also the structure of diamond 
(with carbon atoms at these positions instead 
of silicon). 

examples of important shapes whose key features could easily be taught 
in our schools. 

The silicon chip, which has transformed the industrialized world in 
just a few decades, is based on a structure that is a carrier of incredibly 
miniaturized circuits. Although the circuits themselves are complex, 
the crystal structure of the silicon that houses them is a simple modular 

For example, crystalline silicon is built of linked zig-zag hexagonal 
rings (Figure 31), which are easy to make and instructive to study. In 
the silicon structure the rings are linked to form cage-like polyhedra. 
Elementary school children can learn to build and identify these sub- 
structures, middle school children can learn to put them together, and 
high school children can study the rdation between the silicon structure 
and the properties that make it so useful. 

The CAT scan and other forms of computer-assisted image recon- 
struction have revolutionized medical diagnosis in recent years. While 
diagnosis by X-ray is an exercise in reading shadows, diagnosis by CAT 
scan is an exercise in reconstructing images from their cross sections. 
Like the circuitry on a silicon chip, the image reconstruction used in this 
technology is a complex process, but the simplest geometrical principles 
that underlie it are easily understood. 

Here again we find that the same geometric principles are central to 
many fields. For example, the construction of shapes from sections and 
shadows has been the task of architects and builders for centuries. While 
it is not feasible to bring a CAT scan machine or a construction site 
into the classroom, many projects suitable for school can help students 
understand the relation between shadow or cross section and shape. 

Snowflakes, especially the feathery ones, are enchanting. Children of- 
ten learn to make paper snowflakes in school, an exercise that can easily 



FIGURE 32. Branched snowflakes reveal the familiar hexagonal 
symmetry of ice crystals repeated fractal-like at every scale. 

be extended to a study of their symmetry. The hexagonal symmetry of 
the snowflake provides an introduction to the symmetry of polygons; it 
is an ideal subject for the elementary classroom. 

But the snowflake has much more to teach us. In the first place a 
snowflake looks like a pattern we might see in a kaleidoscope, and so it 
is. This suggests a study of the kaleidoscope, which, as we have seen, 
is an application of the principles of mirror geometry. These same 
reflection principles undergird contemporary technology: one need only 
think of the reflection beams of burglar alarms and lasers or of radar 
and sonar. Middle school children can easily understand and appreciate 
such applications. At the high school level the emergence of hexagonal 
symmetry from aggregates of water molecules can be explored and so 
can the crystals' dendritic growth, or branching. 

The branching of the snowflake is as characteristic as its symme- 
try and is equally significant in the study of shape. First, corners of 
the snowflake sprout beyond a hexagonal "core." Then these branches 
themselves sprout branches, the branches of the branches branch, and so 
forth (Figure 32). The result is a structure in which a certain feature 
branching is increasingly repeated on a smaller and smaller scale. If 
this process could be repeated indefinitely, the result would be a self- 
similar structure; indeed, the snowflake is a fractal at an early stage of 
its development. 

SHAPE 175 


The role of geometry is a perennial issue in mathematics education 
at all levels from elementary school to graduate school For many years 
geometry has been the problem child of the mathematics curriculum. 
A glance through the National Council of Teachers of Mathematics' 
1987 Yearbook Learning and Teaching Geometry 14 suggests some of 
the many questions involved. The problem with geometry is due in part 
to lack of agreement on what geometry is and why we should study it. 
Do we study it to learn disciplined thought? To prepare students for 
other subjects? Or is it because there is important content in the subject 

Most high school geometry texts do point out examples of geometric 
forms in nature, science, technology, and art, although none of these 
connections is ever explored in any depth. The synthesis of method 
and content is usually unsuccessful. Our geometry courses are uneasy 
compromises between many important but very different goals: teaching 
deductive reasoning, proving theorems of Euclid, introducing problem 
solving, teaching visualization, and preparing the students for calculus. 
The continuing debate indicates that none of these goals is particularly 
well served in the present situation. 

In almost all of these debates the teaching of geometry is defended 
on the grounds that it serves external purposes, rather than on the im- 
portance of the subject in its own right. For example, a recent article 
on similarity 7 justifies the teaching of similarity with the following ra- 

Similarity ideas are included in many parts of the school curriculum. Some 
models for rational number concepts are based on similarity; thus, part of the 
students' difficulties with rationals may stem from problems with similarity 
ideas. Ratio and proportion are part of the school curriculum from at least the 
seventh grade on, and they present many difficulties to the student. Standardized 
tests include many proportion word problems. Verbal analogies (a:b::c:d) form 
parts of many intelligence tests. Similar geometric shapes would seem to provide 
a helpful mental image for other types of proportion analogy situations. 

All of these reasons are valid ones, but there is a striking omission: the 
principal reason for teaching similarity ought surely to be that it is of 
profound importance in understanding shape. 

Meanwhile, outside the halls of education, the computer revolution is 
rapidly changing the world in which we live. These changes are placing 
new demands on the curriculum, demands that are just beginning to be 
heard in the schools. The revolution in the study of shape and form 
made possible by the computer suggests that what we need is not just a 
better compromise for geometry, but a new and coherent mathematics 
curriculum that integrates shape into the entire course of study. 


Is Euclid to stay or go? This is not a useful question. We need 
to ask instead what we want our students to know and why. Euclid 
realized that careful reasoning about shape requires careful statements 
of definitions and assumptions and very careful argument. In order 
to analyze shapes, students must know how to measure lengths, areas, 
and volumes as well as planar and dihedral angles. They need to know 
properties of parallel and perpendicular lines, basic angle theorems, and 
fundamental properties of figures. Moreover, they need to know how, 
with ruler and compass, to construct such standard figures as equilateral 
triangles, regular hexagons, and squares. 

The study of shape therefore overlaps the traditional geometry cur- 
riculum, but it cannot be subsumed under it as a brief module or 
extracurricular activity. The shapes that students need to understand 
today, and the things that they need to be able to do with them, are 
too vast a subject for that. Moreover, there is considerable difference 
of emphasis and purpose. 

Traditional geometry shares more than just historical roots with clas- 
sical civilization. Its role in school in some respects is analogous to the 
curricular issues of classical versus modern languages. A student who 
studies Latin or Greek learns rigorous thought and some important his- 
tory and also acquires the basis for many modern languages, including 
our own. Modern, spoken languages, on the other hand, are less rigor- 
ous yet more fluid. They are the living flexible languages that people 
actually use in everyday life. Ideally, students should learn both classical 
and modern languages, although few have the time or opportunity for 
both. A watered-down Latin course enlivened with examples of cognate 
words in Italian, Spanish, or French is not the solution to the problem. 

All of the virtues of Latin and Greek are shared by classical Euclidean 
geometry. For over 2000 years Euclid's Elements has served not only as 
the cornerstone of geometry but also as the very model of mathematical 
reasoning. Deductive reasoning from axioms has been very fruitful, not 
only for mathematics but also for science and philosophy. For example, 
it was questions raised by Euclid's axioms rather than observation of 
the real world that led to the discovery of non-Euclidean geometry, 
which subsequently became the central tool in studying the large-scale 
structure of the universe. Classical geometry has not lost its value, but 
other needs require that we also introduce the mathematical counterpart 
of modern language courses into our curriculum. 

Studying Shape 

Shape is a subject that cuts across many parts of mathematics and sci- 
ence. It offers a rich variety of possibilities for imaginative, exploratory 

SHAPE 111 

instruction from building models to using computers, from observa- 
tion to experiment, from manipulation to calculation. Shape holds ex- 
traordinary potential for enhancing the quality of mathematics instruc- 
tion, in several different ways. 

The study of shape is interdisciplinary. As we have already noted, 
many subjects in which shape plays a role are not usually thought of 
as mathematics in a narrow or restricted sense. For example, problems 
of size and scale do not belong exclusively to mathematics. They lead 
to all sorts of questions that send us to the library or to colleagues in 
other departments. Could there ever have been giants? Could there 
be people as small as mice? Our myths show that these questions are 
older than our recorded history. The answers are not straightforward 
applications of similarity. A giant could not be supported by his legs 
if they were exactly similar to our legs; instead, the bone mass has to 
be increased disproportionately. This complication makes the study of 
biological scale more fascinating than it would be if the answers were 

Perspective is taught in art classes; geometrical optics is a branch of 
physics; similarity and other transformations are central concepts of bi- 
ology; chemists build polygons and polyhedra to model the structure of 
molecules. Even within mathematics, shape is interdisciplinary: it re- 
quires visual and computational skills, logical thought, and many other 
tools. Teaching shape in a coherent, meaningful way can stimulate close 
cooperation among teachers of many subjects. 

The study of shape suggests projects cutting across several subjects. 
For example, the study of similarity can be nicely complemented by a 
study of lenses, requiring an excursion into physics. Even the names 
associated with many of the laws of geometric optics (e.g., Fermat's 
principle) stand as testimony to the fact that today's disciplinary borders 
have not always been so high. It is precisely because we find the same 
shapes everywhere that we need to study them as part of mathematics 
in many different contexts. 

The study of shape is a laboratory subject. All of us, children and 
adults, learn about shapes by making them and studying models (Fig- 
ure 33). As an ancient proverb says, "I hear and I forget; I see and I 
remember; I do and I understand." 

If we wish to build a shape a cube, a scale-model house, or a spiky 
star polyhedron we have to be able to cut out and assemble pieces 
of the correct sizes. This is one of the reasons that basic geometry 
(angle measurement, parallel lines, and so forth) remains indispensable. 



FIGURE 33. The eminent geome- 
ter H.S.M. Coxeter studying a model. 
Coxeter has devoted his life to discov : 
ering patterns in shapes. 

Building models, in this very concrete sense, is one of the best ways to 
unify theory and practice. 

Hands-on experimentation is essential For example, when we make 
a cube with our own hands, we gain much more insight into its metric, 
combinatorial, and stability properties than if we just look at one. If 
instead of cardboard squares we make the cube from plastic straws stuck 
in balls of putty or in marshmallows, the cube will wobble. Though less 
elegant, the wobbly cube is not a "bad" model. On the contrary, it is 
a useful one because it teaches something about rigidity and flexibility. 
It also teaches something about the shapes into which the cube can be 
transformed while maintaining its combinatorial structure. 

Everyone seems to agree that models and "manipulatives" are valuable 
tools in the classroom. But too often one hears the lament that "if 
only models were introduced early enough, we wouldn't have to use 
them later on." This unfortunate attitude masks two implicit but very 
inaccurate assumptions. First, that gross morphological shape is the 
main thing that we learn from models and that it can all be learned in 
elementary school: if you've seen one cube (once), you've seen them all. 
This, of course, is nonsense: the humble cube plays a key role in the 
study of volume, congruence, symmetry, and modular structures. 

The second assumption is that the main purpose in studying a model 
is to develop our powers of abstract reasoning; here the model plays the 

SHAPE 179 

role of training wheels on a bicycle. Certainly we want our students to 
understand the sense in which a particular cube represents the general 
concept of a cube. But even once this is understood, most of us still 
have a lot to learn from real models. 

Ideally, shape should be taught in a laboratory setting. At the very 
least, every school should have a laboratory where students can explore 
shape. A shape laboratory should include work tables, drawing and con- 
struction equipment, three-dimensional models of many kinds, materi- 
als for building them, and places to display them. If possible, it should 
include computers with graphics capabilities. Textbooks should be sup- 
plemented with workbooks, project material, and interactive computer 
graphics programs. 

The study of shape is for everyone. It is often said that studying shape 
is ideal for slow learners. Certainly it is true that students who have trou- 
ble with axioms and abstractions will find a hands-on, problem-oriented 
shape curriculum less difficult and more meaningful. The misconcep- 
tion lies on the other side of the coin the widespread belief that more 
advanced students do not need to study shape. 

We do not have to look further than today's newspaper for evidence 
of the folly of this belief. "Supercomputer Pictures Solve the Once 
Insoluble," proclaimed the headline of a recent article on the front page 
of the New York Times. 13 

Scientists who are using the new supercomputer graphics say that by viewing 
images instead of numbers, a fundamental change in the way researchers think 
and work is occurring. "The human brain is the best pattern recognizer in 
history," says Heinz-Karl Winkler, a Los Alamos National Laboratory physicist. 
"We can use it to visually scan vast quantities of data. We can zero in on a 
structure in an image and distinguish between important things and unimportant 

It is our best students, not our weakest ones, who will be using super- 
computers to study the shape of data and scientific images. How will 
they know how to distinguish important from unimportant things in a 
structure if they have never studied structure at all? 

The study of shape is fun. Students enjoy working with shape, as we 
all do. In teaching shape, especially in a workshop setting, a teacher 
is unlikely to encounter the lack of motivation or the resistance that 
sometimes arise in geometry courses. Unfortunately, fun is suspect in 
some educational circles. One effective way to answer questions about 
the educational value of exploring shape is to hold an open house in 
the shape laboratory so that doubters can become converts by getting 
involved with the material themselves. 


The study of shape is open ended. In a time of rapid change the study 
of shape facilitates open ended strategies for learning. For example, 
computer graphics is revolutionizing the study of shape. Just as the 
supercomputer is changing methods of research, so ordinary computers 
are providing images that most of us could not imagine a decade ago. 

Many teachers say that computer software has completely changed 
the way they teach. They no longer feel that they have to have all the 
answers; instead, they become partners with the students in exploring 
the properties of shape. These teachers are very enthusiastic about their 
new way of teaching. Both their enthusiasm and the new "partnership 
pedagogy" can be encouraged by imaginative curricula that embed ex- 
ploration of shape throughout the entire curriculum. 


1 . Arnheim, R. Visual Thinking. Berkeley, CA: University of California Press, 1 969. 

2. von Baeyer, H.C. "A dream come true." The Sciences (New York Academy of Sci- 
ences), (Jan.-Feb. 1989), 6-8. 

3. Berger, Marcel. Geometry. New York, NY: Springer- Verlag, 1987. 

4. Coxeter, H.S.M. Introduction to Geometry. New York, NY: John Wiley & Sons, 1 969. 

5. Coxeter, H.S.M. Regular Polytopes. New York, NY: Dover, 1973. 

6. Edgerton, Samuel Y., Jr. "Galileo, Florentine 'Disegno/ and the 'Strange Spotted- 
nesse' of the Moon/' Art Journal, 44 (1984), 225-232. 

7. Friedlander, Alex and Lappan, Glenda. "Similarity: Investigations at the Middle 
Grade Level." Learning and Teaching Geometry, K-12. Reston, VA: National Coun- 
cil of Teachers of Mathematics, 1987, 136-143. 

8. Galileo. Sidereus Nuncius (The Starry Messenger), 1610. 

9. Gombrich, E.H. The Sense of Order. Ithaca, NY: Cornell University Press, 1979. 

10. Griinbaum, Branko and Shephard, G.S. Tilings and Patterns. New York, NY: W.H. 
Freeman, 1987. 

11. Griinbaum, Branko. "Geometry strikes again." Mathematics Magazine, 58:1 (1985), 

12. Holden, Alan. Orderly Tangles. New York, NY: Columbia University Press, 1983. 

13. Markoff, John. "Supercomputer pictures solve the once insoluble." The New York 
Times (Oct. 30, 1988), 1, 26. 

14. National Council of Teachers of Mathematics. Learning and Teaching Geometry, 
K-12. Reston, VA: National Council of Teachers of Mathematics, 1987. 

15. National Council of Teachers of Mathematics. Curriculum and Evaluation Standards 
for School Mathematics. Reston, VA: National Council of Teachers of Mathematics, 


16. Senechal, Marjorie and Fleck, George. Patterns of Symmetry. Arnherst, MA: Univer- 
sity of Massachusetts Press, 1977. 

17. Senechal, Marjorie and Fleck, George. Shaping Space: A Polyhedral Approach. 
Boston, MA: Birkhauser, 1988. 

1 8. Senechal, Marjorie and Fleck, George. The Workbook of Common Geometry. (In 

19. Senechal, Marjorie. "Symmetry revisited." In Hargittai, Istvan (Ed.): Symmetry II. 
Elmsford, NY: Pergamon Press, 1989. 

SHAPE 181 

20. Stevens, P. Patterns in Nature. Boston, MA: Little, Brown & Company, 1974. 

2 1 . Thompson, D' Arcy W. On Growth and Form, Abridged Edition. Cambridge, MA: 
Cambridge University Press, 1966. 

22. Watson, James and Crick, Francis. "Structure of small viruses." Nature, 111 (1956), 

23. Weeks, Jeffrey R. The Shape of Space. New York, NY: Marcel Dekker, 1985. 



Every natural phenomenon, from the quantum vibrations of sub- 
atomic particles to the universe itself, is a manifestation of change. 
Developing organisms change as they grow. Populations of living crea- 
tures, from viruses to whales, vary from day to day or from year to 
year. Our history, politics, economics, and climate are subject to con- 
stant, and often baffling, changes. 

Some changes are simple: the cycle of the seasons, the ebb and flow 
of the tides. Others seem more complicated: economic recessions, out- 
breaks of disease, the weather. All kinds of changes influence our lives. 

It is of the greatest importance that we should understand and control 
the changing world in which we live. To do this effectively we must 
become sensitive to the patterns of change, including the discovery of 
hidden patterns in events that at first sight appear patternless. To do 
this we need to: 

Represent changes in a comprehensible form, 

Understand the fundamental types of change, 

Recognize particular types of changes when they occur, 

Apply these techniques to the outside world, and 

Control a changing universe to our best advantage. 

The most effective medium for performing these tasks is mathematics. 
With mathematics we build model universes and take them apart to see 
how they tick, we highlight their important structural features, and we 
perceive and develop general principles. Mathematics is the ultimate 


In "technology transfer": patterns perceived In a single example can be 
applied across the entire spectrum of science and business. 


The traditional approach to the mathematics of change can be summed 
up in one word: calculus. In calculus the changing system is mod- 
eled by a special equation (technically, a differential equation) that de- 
scribes the relation between the rates of change of different variables. As 
much heavy machinery (both theoretical and numerical) as is required 
is brought to bear in an effort to solve the equation. Preparing students 
for the study of calculus has been the central goal of school mathemat- 
ics; setting up and solving the equations of calculus is the lifeblood of 
traditional engineering mathematics. 

Calculus remains an essential component of the mathematics of 
change. Newer methods such as discrete mathematics and computation 
enhance rather than replace calculus. But mathematics is itself subject 
to change. New problems and new discoveries imply the need for a 
much more varied range of mental equipment. Two important trends 
are worth mentioning: the use of increasingly sophisticated approximate 
methods and exploitation of geometry and computer graphics. The first 
has been made possible by the enormous increase in computer power. 
Because computing is based on digital manipulation, it requires an un- 
derstanding of the discrete as well as the continuous and above all, of 
the relation between the two. 

The second trend is a major triumph of mathematical imagination: 
the use of visual imagery to condense a large quantity of information 
into a single comprehensible picture. Computer graphics has led to the 
discovery that many aspects of change are manifestations of a relatively 
small number of fundamental geometric forms. Mathematicians are 
just beginning to understand these basic building blocks of change and 
to analyze how they combine. The methodology involved has a very 
different spirit from traditional modeling with differential equations: 
it is more like chemistry than calculus, requiring careful counterpoint 
between analysis and synthesis. 

The graphical representation of various mathematical concepts arising 
in the study of change has led to the discovery of a variety of intricate 
shapes, each of which appears in many different dynamical situations 
and is thus a "universal" object in the mathematics of change. 14 Fig- 
ure 1 portrays a number of these shapes. They illustrate well the vast 
differences between today's visual methods and the forms traditionally 



FIGURE 1. New scenery in the landscape of change: (a) period-doubling cascade, 
(b) Lorenz attractor, (c) Ueda attractor, (d) Rossler attractor, (e) vague attractor of 
Kolmogorov, (f) Mandelbrot set. 

studied in geometry, such as triangles and parallelograms. 1 ' 17 Geometry 
is now organic and visual rather than limited and formal. 

In consequence, there are very few branches of mathematics today 
that do not have some bearing on change. In part this is because mathe- 
matics is a highly integrated and interconnected structure. Furthermore, 
change is such a complex and varied phenomenon that we need all the 
ideas we can muster to handle it. To study change the scientist of the 
future will need to combine, in a single integrated world view, aspects 
of traditional mathematics, modern mathematics, experimentation, and 
computation. We will need scientists who reach as readily for a pencil as 
for a computer terminal, who can draw crude but informative sketches 
as readily as a computer graphic, and who think in pictures as readily 
as in numbers or formulas. The entire point of view the mental tool 
kit of the working scientist will be very different from what it was even 
a decade ago. 

The patterns of change in nature and in mathematics are uncon- 
strained by conventional categories of thought. In order to make 
progress we must respond imaginatively and sensitively to new types 
of pattern. Our own patterns of thought must themselves change. 


Variety of Styles 

As the twentieth century draws to a close, a new style of mathemat- 
ics is emerging a style whose characteristic is variety. Mathematics 
is once again developing in close conjunction with its applications to 
science physical, biological, behavioral, and social. Much mathemat- 
ics is inspired by computer or laboratory experiments or by the forms of 
natural phenomena. Conversely, mathematical ideas developed for their 
own sake, or in some distinct area of application, are being transferred 
to other tasks and put to work. 10 ' 25 This variety is a strength of the new 
style of mathematics, and it should be encouraged at all levels. More- 
over, computers (especially computer graphics) allow nonspecialists 
from school children to managers, from school teachers to scientists 
to witness the beauty and complexity of mathematics and to put it to 
work. 3 ' 17 

The emergence of this new style of mathematics does not imply that 
the traditional emphasis on precise formulation of concepts and rig- 
orous logical proof can be abandoned. On the contrary, they remain 
an essential component of the mathematical endeavor. Rigor and pre- 
cision are as essential to mathematics as experiment is to the rest of 
science, and for much the same reason: they provide firm reasons for 
believing that ideas and methods are sound. They are part of the sub- 
ject's internal checks and balances, a constant safeguard against error. 
The training of professional mathematicians will necessarily continue 
to require accurate logical thinking and a precise understanding of the 
meaning of "proof." The use of computers as "experimental tools" in 
mathematics can stimulate and motivate new ideas and problems, but 
these experiments alone cannot provide understanding of why the ob- 
served phenomena happen. Their role is to offer a degree of confidence 
that certain phenomena do indeed occur. 

In fact an important trend has become very noticeable, as experience 
in the use of computers has developed. It is the disappearance of the 
dismissive attitude, "Put it on the computer and that will answer all 
your questions." When the answer to a problem is, say, a single number, 
such as the failure load of an engineering structure, all of one's problems 
indeed do disappear once that number is known. But today a typical 
computer-based investigation may produce several hundred diagrams 
representing the behavior of the system under various conditions. For 
example, think of the flow of air past a space shuttle for different speeds, 
angles of attack, and atmospheric densities. Such a catalogue, despite 
its apparently large size, is likely to be inadequate for determining the 
behavior under all possible conditions. If the system involves three 
adjustable parameters, as does the one just mentioned, and each can take 


up to ten values, then a total of a thousand combinations is possible. 
With four such variables there are ten thousand, with six there are a 

In practice, six is a small number of parameters: simple problems 
in chemical engineering typically involve several dozen parameters and 
may involve hundreds. It is pointless to produce a computerized cat- 
alogue of one million diagrams, let alone a billion or a trillion. The 
fundamental question "What is really going on here?" returns from 
computer science to the realm of mathematics. Such questions require 
input from the human brain far more than from the computer. 

However, the role of the computer should not be underestimated. It is 
becoming an ever more prevalent thinking aid. Computers cannot only 
generate "results," but they can also be used to experiment at interme- 
diate stages of understanding, to test hypotheses and possible mecha- 
nisms. With appropriate safeguards, computer calculations can actually 
produce rigorous proofs of mathematical results. Such computer-aided 
proofs require very careful construction and a great deal of human input 
to set them up: they are far from routine and usually require specially 
constructed software and lengthy machine time. More than anything 
else, they constitute a difficult specialist area of mathematics. "Put it 
on the computer" is no panacea. 

Approaches to Teaching 

For reasons of exposition only, rigorous proof does not feature promi- 
nently in this essay. It is part of the mathematician's basic technique, 
and it remains just as important as it ever was, but it holds much less 
interest for the nonspecialist. Accordingly, its role has not been made 
explicit, although it underpins everything discussed. 

However, the fact that proof is important for the professional math- 
ematician does not imply that the teaching of mathematics to a given 
audience must be limited to ideas whose proofs are accessible to that 
audience. Such a limitation is likely to make mathematics dull, dry, and 
dreary, for many of the most stimulating and exciting ideas depend upon 
highly complex theories for their proofs. Many mathematical concepts 
can be grasped without being exposed to their formal proofs. Using an 
idea is quite different from developing it. It is possible to "explain" quite 
advanced concepts to children by means of examples and experiments, 
even when a formal proof is too difficult. 

For example, in the theory of chaos an important concept is that 
of "sensitivity to initial conditions." If a system evolves from two 
very similar initial states, the resulting motions can quickly become 
totally different. Given access to suitable software, virtually anyone can 


appreciate this sensitive and paradoxical behavior in, say, the Lorenz 
attractor (Figure Ib) merely by watching how two almost equal starting 
values move apart and become independent. However, a rigorous proof 
that the Lorenz system really does behave in the manner that computer 
experiments suggest is not only beyond the capacities of the average per- 
son, it has not yet been achieved by professional mathematicians and 
remains an active problem for future research. 

The breadth of viewpoint and range of skills demanded by today's 
mathematics will be important, not just for mathematicians and scien- 
tists but for people in all walks of life. Change affects us all. Managers, 
politicians, business leaders, and other decision makers must cope with 
a changing world. They must appreciate how subtle change is; they must 
unlearn outdated assumptions. 

It is a tremendous challenge to devise methods of educating a genera- 
tion of such versatile people. Our aim here is to Suggest ways to develop 
in children some of the underlying ideas and to stimulate a new point of 
view. We must advance beyond the traditional approach of arithmetic 
leading to algebra and thence to calculus. 

In the design of an effective new curriculum, one important compo- 
nent is an understanding of the new viewpoints that are developing at 
the frontiers of research. Yet the curriculum must be suitable for all 
children, not just for those who will become research scientists. Nev- 
ertheless, new kinds of mathematics that are evolving at the research 
level set the style for applications and education in the future. Thus it 
is important for teachers and educators at all levels to understand the 
general nature of these new methods and the kinds of questions that 
they address. 

Levels of Description 

The mathematics of change can be viewed at many levels: 

The big picture: What are the possible types of change? 

Specific areas of mathematical technique: How are the equations 

General areas of application: How does the size of an animal 
population vary with time? 

Individual applications: Design a chemical reactor to produce 

Simple theoretical examples: How does a pendulum oscillate? 

Mathematicians operate on all of these levels because insights obtained 
at one level are often transferred to other levels. In mathematical 
technology transfer, patterns are not tied to any particular area of ap- 


Simple theoretical examples are seldom of direct relevance to indus- 
trial applications. For example, an analysis of pendulum dynamics is 
of no direct use in the study of wing flutter in supersonic aircraft. In 
practical terms the pendulum went out with the grandfather clock. But 
simple examples have their uses: they prepare us for the complexities 
of real life. A pendulum makes many important features of oscillation 
more accessible than would a realistic model of a vibrating airplane 

To illustrate these themes we will use some specific questions that 
exemplify the new style of mathematics. These questions have been 
chosen not as specific goals in themselves, but because they motivate 
compelling mathematical ideas: 

How do living populations change? 

Where do meteorites come from? 

Why are tigers striped? 

Only the first of these questions appears to involve change. The oth- 
ers seem to be about static phenomena. 27 Meteorites are just there or 
not at random. A tiger is striped, a leopard is not, and never the 
twain shall meet. In fact the questions are all about change of some 
kind. Do meteorites really plunge into the earth's atmosphere "at ran- 
dom," or does something more structured lie behind their appearance 
in the night sky? A mature striped tiger does not just exist as a static 
object: it develops from a single (unstriped) cell. Somewhere along the 
line of development the stripes first make their entrance. Change is the 
common theme behind each of these varied questions. 


If we put a few rabbits on an uninhabited island, pretty soon there will 
be a lot more rabbits. On the other hand, the growth cannot continue 
unchecked, or soon there would be more rabbits than island. It follows 
that change in a population is affected by both internal and external 
factors. How they combine to influence changes in the population is a 
good example of mathematical modeling that can be studied at many 
different levels. 

Limits to Growth 

We begin with the simplest case: a population consisting of a single 
species with a constant (and therefore limited) food supply. Figure 2 
shows typical experimental data for growth of such a population. Its 
typical S-shaped curve is characteristic of many growth phenomena. 27 


FIGURE 2. Changes in the size 
of a yeast population growing in 
an environment with a limited 
food supply. Time 

Similar curves arise if we measure particular features of a single devel- 
oping organism for example, the height or weight of a growing child. 

It is common in many families to record the heights or weights of 
children as they grow. These charts may be displayed on classroom 
walls for discussion and comparison. The growth of young children 
in a single class over a period of one or two years will illustrate linear 
growth. The heights on the chart, plotted against time, will lie close to 
a straight line. However, the complete growth record of a child from 
birth to adulthood exhibits the characteristic S shape. Neither the initial 
phase nor the final phase is linear. Early on the growth is approximately 
exponential; later it saturates as it approaches a constant value. 

Children who have recorded growth curves can be introduced to the 
entire 5-shaped curve, either as an experimental observation or as a 
table of numbers. A good exercise for children in middle school is to 
use evidence from several childrens' growth curves together with data 
from their own childhood to project their own adult heights. Later, 
as older students, they can learn how to represent these curves with 
formulas. Children can be encouraged to analyze the main features of 
this curve and to consider why the curve has them. 

Suppose Alice is 1 foot tall at age and 4 feet tall at age 8. If this 
growth rate continues 3 feet every 8 years how tall will she be at 
ages 16 or 24 or 32? (Answers: 7 feet, 10 feet, 13 feet.) Even young 
children can see that these answers are not credible. What's wrong? The 
mathematics is fine, but the model linear growth is inappropriate. 
Moral: When you use mathematics you have to pick a sensible model 
and not just calculate numbers blindly. 

Levels of Analysis 

A study of population growth can be carried on at several levels 
verbal, numerical, graphical, dynamical with the sophistication in- 
creasing as the children become older. Verbal description of the yeast 
growth curve shows a population that increases slowly at first but then 
grows exponentially. That is, the breeding population increases by a 


constant factor in successive periods of time. However, when the pop- 
ulation becomes sufficiently large, the rate of growth slows down, even- 
tually leveling off at a steady maximum value. 

This verbal model is purely descriptive. It is mute about why the 
population levels off. The verbal description is helpful for general intu- 
ition but useless for further analysis of behavior. Its principal role is to 
summarize simply the pattern of growth. 

To appreciate the effect of exponential growth and to gain insight 
into why such growth cannot continue unchecked children can be told 
the famous story about the emperor's reward. In a far country a person 
performed an important task for the emperor, and she was asked to 
name her reward. The reply was: "One grain of wheat on the first square 
of a chessboard, two on the next, then four on the next, then eight, 
and so on, doubling each time." The emperor was not very impressed 
. . . until he worked out how the numbers grew! 

Children can do the same with a calculator or a computer. Younger 
children can experience exponential phenomena without using large 
numbers by folding a sheet of paper repeatedly in half. How many 
times can you manage before you get stuck? 

Data on weights of animals, wingspans of birds, girths of trees, num- 
bers of leaves on plants, etc., can be gathered (or presented) in the form 
of numerical tables. Children can look for patterns in the numbers: Are 
they increasing? Decreasing? Constant? They can calculate differences 
and ratios, make tables, and look for patterns. Numerical tables lead 
naturally to graphical representation. 

The growth curve provides a visual picture of the way in which the two 
variables, population and time, are related. Such a graph, sometimes 
called a time series, replaces numerical information by graphical: it is 
the simplest example of the geometrization of change. The idea that 
numbers can be represented by the positions of points, and changing 
numbers by curves, is the basis of all geometric methods in the mathe- 
matics of change (see Figure 3). Children need many opportunities to 
learn that in mathematics a picture is indeed worth a thousand words. 

For younger children experimental work is most appropriate. They 
can count the number of eggs produced by ducks or chickens, measure 
the height of a growing plant, measure the temperature each day at 
noon, record the position of the moon in the sky. By graphing this 
data, children can search for patterns of change and discuss possible 

Older children can be set more ambitious tasks: the water level in 
a pond, the number of leaves on a bush, the movements of the stock 
market, experiments from physics and chemistry laboratories. Using 
data from real phenomena is an effective way to integrate mathematics 
into other school subjects. Algebra students can also use mathematical 


















Square Wave 




FIGURE 3. Some of the many different types of change, together 
with their typical time series. 

processes and formulas to generate theoretical data, to look for patterns, 
and to compare theory with reality. 

Dynamical Systems 

The next level of exploration is to model not the patterns in the num- 
bers but the process that gives rise to these patterns. In the traditional 


approach this idea leads to differential equations and thus requires 
calculus. But another possibility increasingly attractive in an age of 
computers is to throw off the chains of calculus and take seriously the 
fact that the number of creatures in a population is discrete rather than 

Imagine that time t increases in. discrete whole number steps, t = 1, 
2, 3, .... The value of the population p at time t is written as p(t). 
Its next value p(t+l) can then be related to its current value p(t) by a 
specific growth law. This type of model is called a difference equation 
or a discrete system. 7 ' 28 

In living populations, unchecked breeding at a constant rate m cor- 
responds to a law of the form p(t + 1) = mp(t\ leading to exponential 
growth: p(t) = p(Q)m' 9 where p(0) is the initial population. The law of 
restrained growth, which allows for limits imposed by lack of food or 
space, modifies this law by subtracting a correction factor that reflects 
these limits: 

p(t+l) = mp(t)-n\p(t)] 2 , 

where m and n are constants that depend on the particular circum- 
stances. This equation, known as the Verhulst law (named after the 
nineteenth-century French scientist P.P. Verhulst), is one of the most 
common algebraic models of limited growth. 

Students can study this equation with tools from simple algebra, both 
by making tables and by simplifying the equation. The population level 
p(t) = m/n is a cutoff level: once it is reached, the next value, p(t+l),is 
0, as are all subsequent values. To study how the population compares 
with the cutoff level, we can express p(f) as a proportion of m/n by 
changing the units of measurement by letting p(t) = q(t) (m/n). This 
leads to the equation 

where q(t) expresses the population as a fraction of the cutoff level. 
Instead of two parameters m and n, we now have just one parame- 
ter m, which makes the mathematics much simpler. Because q(t) is a 
proportion of the cutoff population, it will be some fraction between 
and 1. 

Difference equations such as the Verhulst law are ideal for computer 
calculation, because they express a simple repetitive procedure for de- 
scribing the behavior at the next instant 1 + 1 from the behavior now, at 
time t. With a computer we can easily calculate solutions of the discrete 
Verhulst law without knowing a formula for these solutions. 


(Indeed, there is no general formula for these solutions.) We can then 
encapsulate the results in a single geometric object such as a time series 

This illustrates an important general principle: discrete mathematics 
is often more accessible than continuous mathematics (calculus). The 
Verhulst law can be introduced and studied through tables of values as 
soon as students begin their study of algebra, usually four years before 
they are introduced to calculus. However, it is also harder to derive the 
detailed mathematical structure of discrete systems, and their treatment 
tends to be experimental or at least computer based. 

Numerical Experiments 

The Verhulst law offers an excellent opportunity for numerical ex- 
periments using only elementary arithmetic and calculators. 3 ' 23 Even 
elementary school children can follow the rules, years before they are 
introduced to the formalism of algebra. The Verhulst law, whose alge- 
braic form is 

p(t + 1) = m\p(t) - p(t) 2 ] 

can easily be translated into a table or a spreadsheet for exploration for 
various values of the parameter m. (Note that we are now using p to 
signify the population proportion, which we previously called #, rather 
than the population size.) 

Start with some value of p(0), say 0.1, and calculate in turn p(l), 
p(2), p(3), .... In words: new population equals old population minus 
the square of the old population, multiplied by a constant. 

For example, suppose m 2. Then the successive values are 

0.1, 0.18, 0.295, 0.416, 0.486, 0.499, 0.5, 0.5, ... . 

We see initial growth, settling down to a specific final level. This growth 
is similar to the experimentally verified growth of yeast and other ho- 
mogeneous populations (see Figure 2). When m = 3 we get 

0.1, 0.27, 0.591, 0.725, 0.598, 0.721, 0.603, 0.717, .... 

The values in this case appears to oscillate between about 0.6 and 0.7. 
(In fact, this oscillation eventually dies out, but very slowly: it becomes 
more apparent at m = 3.1 or 3.2.) Finally, consider m = 4: 

0.1, 0.36, 0.922, 0.289, 0.821, 0.585, 0.970, 0.113, .... 

Now we see no clear pattern at all! What has happened? 

The Verhulst law leads to a rich range of behavior, including periodic 
oscillations and apparently patternless, irregular behavior. The latter 
is known as chaos. Here a simple experiment using a calculator brings 



quite young children to the frontiers of research. Indeed, this example 
can lead to an enormous range of classroom activities: working out 
numerical values on calculators, computers, or electronic spreadsheets; 
graphing the results; spotting patterns; analyzing why they occur. 

Traditionally, random-looking behavior is modeled by statistics, using 
equations that incorporate explicit random terms. But there is no ran- 
dom term in the Verhulst law: it is deterministic. This example shows, 
surprisingly, that behavior predicted by a simple and explicit law can 
be highly irregular, even random. 

This paradoxical discovery is called deterministic chaos. Irregular fluc- 
tuations may arise from nonrandom laws, making it possible to model 
many irregular phenomena in a simple manner. It also demonstrates 
that simple causes can produce complicated effects. It is one of the 
most exciting areas of current mathematical research. 5 ' 11 ' 24 

The Irregular Fruit Fly 

It is always possible that chaotic behavior could be just an artifact 
of the model and not a phenomenon of nature. Perhaps. But natural 
populations do, in fact, display irregular oscillatory behavior. Figure 4 
shows experimental data on a population of fruit flies kept in a closed 
container and fed a constant protein diet. 15 When the population rises 
too high, there is too little food and the flies are unable to breed properly. 
The population then drops until there is excess food; then the flies breed 
unrestrictedly and the population shoots up again. 

The main overall effect is an oscillation with a period of about 38 
days. However, as the time series shows, the way in which the popula- 
tion changes is decidedly complex. Many of the peaks in the graph are 
double, being more M-shaped than A-shaped. The height of the peak 



300 400 





FIGURE 4. Variations in an experimental population of fruit flies show irregular 
oscillatory behavior that is typical of deterministic chaos. 


varies: small, medium, large, in turn. After the first 450 days or so, the 
changes become more and more irregular. 

This graph illustrates an important question for mathematical model- 
ing and for the analysis of scientific data. Some of the observed changes 
are due to the population dynamics of fruit flies. Others may be due to 
outside effects such as contaminated food, disease, or for all anyone 
knows the tides or the position of Mars in the sky. How can we tell 
which are which? 

It would be easy to assume that the regular effects the M-shaped 
peaks, the modulation in their size are unrelated to outside causes but 
that the increasing irregularity after 450 days is due to something go- 
ing wrong with an outside cause. However, this assumption may be 
incorrect. Numerical experiments with models similar to the Verhulst 
law show that simple mathematical laws can produce both regular os- 
cillations and irregular chaos, just by making slight changes to a single 
parameter. In fact many aspects of the fruit fly data, irregularities in- 
cluded, can be modeled by simple systems. 

Children can be brought to understand the possibilities for complex 
behavior in simple systems by performing numerical experiments, first 
with calculators and later with computers. They can then search for 
patterns in apparently irregular data. For example, given a time series 
generated by the Verhulst law or related equations, they can plot p(t+l) 
against p(t) and observe that all the points lie on a smooth curve. They 
can analyze the curve to determine its geometric features; older children 
can seek an appropriate formula and estimate the value of the growth 
rate parameter rn. 

More sophisticated versions of this geometric technique have been 
applied to many sets of observational data, for example, to the appar- 
ently random fluctuations that occur in the numbers of people suffering 
from a disease such as measles. Often the experimental time series 
appear random. But graphical analysis suggests that a simple process, 
resembling a difference equation, underlies the apparent irregularities. 
In consequence it is often possible to set up simple but realistic models 
that reproduce the patterns of change in these systems. 

Moving on to Calculus 

Traditional analysis via calculus still has an important role to play in 
modeling population growth. In this case it provides a formula rather 
than a picture or a list of numbers. In the calculus-based model the 
value of p(t) need not be a whole number, whereas a real population 
necessarily takes on whole number values. The model is thus a con- 
tinuous approximation to a discrete phenomenon. This is a common 


technique and is often used when the maximum population size is fairly 
large. Then the change caused by adding or removing a single individ- 
ual is extremely tiny, so that the possible range of sizes cannot easily 
be distinguished from a continuous range. The resulting model is a dif- 
ferential equation, one of the key concepts of higher mathematics. 4 A 
differential equation involves not just variables such as the population 
p but also rates of change of variables. The rate of change of a variable 
p with respect to time is traditionally denoted by dp/dt. 

The simplest differential equation for populations is a law of uniform 
growth dp/dt = mp. This states that the rate of change dp/dt of the 
population p at a given moment t is proportional to the population p at 
that same moment, where the constant of proportionality is m. In other 
words, a larger population produces proportionately more offspring than 
a smaller one. The solution to this differential equation isp(t) = p(0)e mt 
for an initial value p(0) at t = 0, which is the continuous version of 
exponential growth. The population explodes, unchecked. 

In practice other factors must come into play to limit the growth. As 
with the Verhulst law, we modify the equation by subtracting a term 
np 2 (where n is a second constant): 

dp/dt = mp - np 2 . 

The point of this extra term is that when p is small, p 2 is negligible in 
comparison, so that the correction term np 2 has little effect; in this case 
we obtain (almost) exponential growth. However, as p becomes larger, 
the term -np 2 begins to dominate the dynamics, substantially reducing 
the rate of growth. Indeed when p reaches the value m/n, the rate of 
change of the population, dp/dt, becomes zero. When this happens, no 
further growth takes place. So m/n represents the maximum popula- 
tion. Using techniques of calculus, it is possible to find a formula for 
the solution. The graph of this solution, known as the logistic curve, has 
the same S-shape as the experimental data on yeast (Figure 2). 

The rich variety of behavior steady, periodic, chaotic of the dis- 
crete Verhulst law is absent from its continuous analog, which yields 
only a smooth S-shaped curve. This shows in a particularly convinc- 
ing manner that changing from discrete models to continuous ones, or 
conversely, can lead to new phenomena: it is not just a harmless trick. 
Examples such as these raise important questions about the relation 
between continuous and discrete models, relations worth exploring in 
mathematics classes at many school levels. 

The continuous model permits experimental data to be fitted to a the- 
oretical curve, and this opens the way to prediction of future behavior. 
For example, if a logistic curve is fitted to the population of the United 
States up to 1 930, it predicts that by the year 2000 the population should 


level off at around 200 million. More accurate techniques give a pro- 
jected population for the year 2000 of 260 million, about 30% higher. 
So the simplified approach does surprisingly well. Students armed with 
population data (of an ecosystem, a nation, or the world) can try fitting 
the logistic curve to this data to determine the constants m and n and 
to predict future trends. 


The behavior of meteorites is a small part of the general problem of 
the dynamics of celestial bodies of moons, planets, stars, galaxies. The 
regularities, or almost regularities, of the motions of the planets have 
throughout history been a major motivation for the study of change. It 
is not just a matter of fascination with the night sky: important down- 
to-earth problems such as agriculture and navigation have at various 
times depended upon knowledge of the movements of the stars and 

Astronomy is a rich area for finding good classroom activities about 
change: the phases of the moon, the tides, the apparent motion of stars, 
the changing seasons, earth satellites. Another possibility is to recon- 
struct Galileo's experiments using balls on inclined slopes and deduce 
the law of motion in a uniform gravitational field. Data gathered in 
such enterprises can fuel many rich mathematical explorations. 

Historically, our understanding of such matters went through sev- 
eral stages informal description, empirical models, geometrical mod- 
els, dynamical models before culminating in the laws of motion dis- 
covered by Isaac Newton. But these laws often lead to equations that 
are very hard to solve. They can be solved exactly for a system of two 
bodies, where they predict elliptical orbits. The problem of celestial 
motion for a system of three bodies has been notorious for over two 
centuries for its apparent intractability. With modern computers we 
can see why: even simplified versions for example, where one body 
has negligible mass lead to complex and highly irregular behavior. 

Computer packages now simulate planetary motion for systems of 
two, three, or more bodies. Children as young as 1 1 or 12 can use these 
packages to experiment with the behavior of the regular elliptical orbits 
of two-body systems and the complicated behavior of three or more 
bodies. By using these packages, they can gain more insight into the 
geometry of planetary motion than Isaac Newton did in a lifetime of 



Modern understanding of planetary motion stems from work of the 
French mathematician Henri Poincare around the turn of the 
century. 9 ' 22 In 1887 King Oscar II of Sweden offered a prize of 2500 
crowns for an answer to a fundamental question in astronomy: Is the 
solar system stable? We see now that Poincare's response was a major 
turning point in the mathematical theory of celestial change. 

Scientists call a system stable if it does not change when perturbed by 
small disturbances. It is unstable if small disturbances tend to become 
magnified, leading to large changes in behavior. For example, a pin 
lying on its side is stable, whereas a pin balanced on its tip is unstable 
since it will always fall over (Figure 5). 

Children can develop sound intuition about the notions of stable and 
unstable systems, and indeed for the typical complexity of dynamical 
systems, by exploring the behavior of various mechanical "executive 
toys" multiple pendulums, interacting magnets, gyroscopes. For ex- 
ample, consider a pendulum with a magnetic bob, arranged to swing 
over the top of a second magnet. If the two magnets have opposite po- 
larity, then the pendulum is stable in its downward position, attracted 
by the lower magnet. But if the polarities are the same, and you try to 
hold the pendulum over the lower magnet, it tries to move away. The 
downward position is now unstable and the child can feel it! 

Experiments of this type, usually carried out in a rather formal way, 
are currently characteristic of physics classes. Less formal experiments 
should be carried out in mathematics classes as well, as an integrated 
part of the development of intuition for change and motion, for stabil- 
ity and chaos. At later stages, after a child's intuition is better devel- 
oped, such experiences can be formalized with appropriate mathemati- 
cal models. 

FIGURE 5. Unstable and stable states of a pin: when balanced 
on its tip, any wiggle will cause a pin to fall, whereas when resting 
on its side, small forces produce only small changes in the position 
of the pin. 


Stability is an extremely important question. An airplane must not 
only fly, but its flight must be stable, or it will drop out of the sky. When 
a car rounds a corner it must not tip over on its side. The solar system 
is a very complicated piece of dynamics. How do we know that the 
motion is stable? Will all the planets continue to move in roughly their 
current orbits? Could Pluto crash into the sun? Could the earth wander 
off into the cold of the outer planets? These are very subtle problems 
whose answers are very difficult to discern. 

Rubber Sheet Dynamics 

Poincare didn't solve King Oscar's problem: it was too hard. But he 
made such a dent in it that he was awarded the prize anyway. To do it 
he invented a new branch of mathematics now called topology. Often 
characterized as "rubber sheet geometry," topology is more properly de- 
fined as the mathematics of continuity, as the study of smooth, gradual 
changes, the science of the unbroken. 8 ' 18 Discontinuities, in contrast, are 
sudden and dramatic places where a tiny change in cause produces an 
enormous change in effect. 

The celestial motion of two bodies a universe consisting only of the 
earth and the sun, say is periodic: it repeats over and over again, once 
every year. (That is the definition of "year.") This periodic behavior 
immediately proves that in such a solar system containing only the 
earth and the sun the earth would not fall into the sun or wander off 
into the outer reaches of infinity; for if it did, it would have to fall into 
the sun every year or wander off to infinity every year. Those aren't 
things you can do more than once, and they didn't happen last year, so 
they never will. In other words, periodicity gives a very useful handle on 
stability. In our real universe bodies will disturb this simple scenario; 
nevertheless, periodicity is still important. 

Under gravity, two bodies behave simply: they both move in elliptical 
orbits about their common center of gravity. Three bodies behave in an 
unbelievably complicated manner, even if the problem is simplified by 
assuming that one has a very small mass compared with the other two. 
More than three bodies can lead to even worse behavior. 

Juggling is an example of stable periodic motion. It is periodic be- 
cause the same actions are performed over and over again; and it must 
be stable since otherwise it wouldn't work. Juggling two bodies is rela- 
tively simple; juggling more quickly becomes very complicated. If one 
teaches children to juggle, they will learn quickly about the complexity 
of dynamical systems. They can analyze the periodic pattern of juggling 
motion. Why is juggling stable? What is the role of hand-eye feedback? 



FIGURE 6. Poincare's geometric approach to 
periodicity: if the state of a system describes a 
closed loop in phase space, the system must be 
periodic and hence stable. 

Poincare grappled with the existence of periodic solutions, and he 
found that they could be detected by a topological method. Suppose 
that at some particular instant of time the system is in some particular 
state and that at a certain time later it is again in the identical state. 
Then it must repeat, over and over again forever, the very motion that 
took it from that state back to itself. Returning just once to a previous 
state, perfect in every detail, is the essence of periodic motion. 

Topology enters when this idea is made geometric. 24 Imagine that the 
state of the system is described by the coordinates of a point in some 
high-dimensional space, which scientists call phase space. As the system 
changes, this point will move, tracing out a curve in phase space. In 
order for the system to return to its initial state, this curve must close up 
into a loop (Figure 6). Stability of the system thus translates to "When 
does a curve form a closed loop?" The question asks nothing about the 
shape or size or position of the loop, merely that it be closed: it's a 
question for topology. Thus the existence of periodic solutions depends 
on topological properties of the curve that represents the changing state 
of the system in phase space. 

Phase space is an abstract mathematical space with many dimensions 
that represent all possible variables that govern the state of a system, 

FIGURE 7. Example of a phase por- 
trait in which different curves repre- 
sent possible evolution of a system 
under different initial conditions. 


which is itself represented as a single point in phase space. As the state 
changes, this point moves, tracing out a curve, or flow line. The picture 
of how these flow lines fit together is called the phase portrait of the 
system. 1 The flow is typically indicated by curved lines, corresponding 
to the time evolution of the coordinates of various initial points (see 
Figure 7). Arrows mark the direction of motion of time. 

Phase Portraits 

Once children have grasped the concept of graphing the changes in 
a single variable, they can be introduced to phase portraits. Instead 
of plotting the value of a single variable against time, in a time series 
they can plot the sequence of values of two different variables in two 
coordinate directions. Such exercises will develop insight into the mul- 
tidimensional geometry of change. For young children these variables 
might be the height and weight of growing animals or the tempera- 
ture and rainfall per day. Older children could consider astronomical 
phenomena such as the positions of the sun and moon, or measure- 
ments made on an electronic circuit, or observations of a pendulum, or 
price movements of two different exchange rates on the world currency 

The oscillations of a simple pendulum provide a very illuminating 
example of a phase portrait but suitable in full detail only for more 
advanced students. The traditional approach to the pendulum is to 
write down an approximate equation whose solution is a sine curve. 
The approximation is necessary because standard techniques of calculus 
cannot solve the true equation for an exact model. The student does 
learn useful properties of the sine curve as well as a formula for the 
period of a pendulum whose swings are small. However, this traditional 
approach is in some respects unsatisfactory since the approximations 
employed are rarely justified. It leaves the unwarranted impression that 
lack of precision is acceptable in mathematics. 

Instead, the law of conservation of energy can be applied to yield an 
exact model for the motion of a pendulum. It leads to the equation 

v = 

where v is velocity, C and k are constants, and 6 is the angle that the 
pendulum makes with the vertical. By sketching this family of curves, 
one in effect draws the phase portrait (Figure 8). All of the motions of a 
real pendulum, including large swings even cases when it revolves like 
a propeller can be seen in this picture. 24 With this alternate approach, 
students obtain equally valid practice with the sine function, an accurate 




FIGURE 8. Phase portrait of a pendulum in which all possible motions are visible. 

model, no approximations, and an important physical principle (conser- 
vation of energy). Isn't that a better way to think about the pendulum? 


The dynamical equations for three bodies cannot be solved by a for- 
mula, but they can be put on a computer and solved numerically. Such 
models provide a good means of exploring the surprising effects of res- 
onance on the motion of dynamical systems. Resonance occurs when 
different periodic motions have periods that are in some simple numer- 
ical relationship such as 1:1, 2:1, 3:2, and so on. For example, Titan, a 
satellite of Saturn, has an orbital period that is close to 4:3 resonance 
with that of another satellite, Hyperion. Specifically, Hyperion takes 
21.26 days to complete one orbit and Titan takes 15.94. The ratio of 
these is 1.3337, convincingly close to the ratio 4:3. 

Older children can use a computer package to simulate planetary dy- 
namics. They can study the motion of the moon or of a satellite in 
transit from earth to moon. They can study the way in which Jupiter's 
satellites are locked into resonant orbits. They can study the so-called 
Lagrange points, where satellites (or space colonies) can remain in sta- 
ble positions 60 ahead of or behind the moon. This too is a kind of 

Resonances are especially important in dynamics. They lead to a rich 
and subtle geometry that is almost unbelievably complex. In Figure 9 
the large circles represent regular motion; secondary "islands" between 
the circles represent resonances; tertiary islands signal more delicate 
multiple resonances. The spaghetti-like crossings represent chaos. The 
structure repeats forever on smaller and smaller scales. 



FIGURE 9. Fractal structure near a periodic orbit: islands signal 
resonances of various orders, while tangles represent regions of 

High school students can easily search astronomical tables to look 
for evidence of resonances. This work involves plenty of practice with 
fractions, decimals, calculators, and computers. It shows how simple 
mathematics can produce deep insights to those who look at the world 
from a mathematical perspective. 

Resonances often generate chaos. Figure 9 has a particular disturbing 
quality of self-similarity: each island has the same complexity, indeed 
the same qualitative form, as the entire picture. This complicated self- 
similar structure is not some mad mathematician's nightmare. It's what 
really happens. 

The concept of self-similarity together with the associated ideas of 
fractal geometry, 14 can be made accessible to children around the age 
of twelve, maybe younger. The topic can be introduced using natu- 
ral examples: coastlines, leaves, ferns, etc. Next, computer models of 
fractals such as the Cantor set and snowflake and dragon curves can 
be drawn and their patterns analyzed. Concepts of fractal structure and 
self-similarity can easily be developed from these examples. Even young 
children can appreciate the idea of fractal dimensions which need not 
be whole numbers. 


Gaps and Clumps 

Resonances feature prominently in another astronomical conundrum, 
the gaps in the asteroid belt, which is directly related to our original 
question about meteorites. Most asteroids circle between the orbits of 
Mars and Jupiter, although a few come much closer to the sun. How- 
ever, the asteroid orbits are not spread uniformly between Mars and 
Jupiter. Their radii tend to cluster around some values and stay away 
from others (Figure 10). Daniel Kirkwood, an American astronomer 
who called attention to this lack of uniformity in about 1860, also no- 
ticed an intriguing feature of the most prominent gaps: if an asteroid 
were to orbit the sun in one of these Kirkwood gaps, then its orbital pe- 
riod would resonate with that of Jupiter. Conclusion: Resonance with 
Jupiter somehow perturbs any bodies in such orbits, causing some kind 
of instability that sweeps them away to distances at which resonance 
no longer occurs. The special role of Jupiter is no surprise since it is 
so massive in comparison with the other planets. The gaps are obvious 
in recent data, especially at resonances 2:1, 3:1, 4:1, 5:2, and 7:2. On 
the other hand, at the 3:2 resonance there is a clump of asteroids, the 
Hilda group. So stability is not just a matter of resonance: it depends 
on the type of resonance. The questions remain a subject of intense 

Recent computer calculations 30 show that an asteroid orbiting at a 
distance that would suffer 3:1 resonance with Jupiter can either follow 
a roughly circular path or a much longer and thinner elliptical path. 
If the orbit of an asteroid is sufficiently elongated, it crosses the orbit 
of Mars. Every time it does so there is a chance that the asteroid will 
come sufficiently close to Mars for its orbit to be severely perturbed. It 
will eventually come too close and be sent off into some totally different 
orbit. The 3:1 Kirkwood gap is there because Mars sweeps it clean, 
rather than being due to some action of Jupiter. What Jupiter does is 
create the resonance that causes the asteroid to become a Mars crosser; 
then Mars kicks it away into the cold and dark. Jupiter creates the 
opening; Mars scores. 

The same mechanism that causes asteroids to be swept up by Mars can 
also cause meteorites to reach the orbit of the earth. The 3:1 resonance 

FIGURE 10. Gaps and clumps in 
the distribution of asteroids reveal res- 
onance with the orbital period of Jupi- 



with Jupiter thus appears to be responsible for transporting meteorites 
from the asteroid belt into earth orbit, to bum up in our planet's at- 
mosphere if they hit it. 26 A cosmic football game, played among the as- 
teroids by Mars and Jupiter, determines whether or not floating cosmic 
rocks and perhaps sometimes mountains will crash into the earth's 
atmosphere. It would be hard to find a more dramatic example of the 
essential unity of the entire solar system or a better example of the 
interconnectedness of change. 


"What immortal hand or eye dare frame thy fearful symmetry?" said 
William Blake, referring to the tiger. Although Blake wasn't using the 
word "symmetry" in a technical sense, it turns out that the behavior of 
symmetric systems has a distinct bearing on the striped nature of tigers. 

Symmetry is basic to our scientific understanding of the universe. 13 
The symmetries of crystals not only classify their shapes but also deter- 
mine many of their properties. Many natural forms from starfish to 
raindrops, from viruses to galaxies have striking symmetries. Man- 
made objects also tend to be symmetric: cylindrical pipes, circular 
plates, square boxes, spherical bowls, hexagonal steel bars. 

That symmetric causes have symmetric effects is a long-standard prin- 
ciple in the folklore of mathematical physics. Pierre Curie made the case 
succinctly: 6 "If certain causes produce certain effects, then the symme- 
tries of the causes reappear in the effects produced." The principle 
seems natural enough but is it true? The question is a subtle one in- 
volving not just the meaning of "symmetry" but also that of "cause" 
and "effect." 

Recently scientists and mathematicians have become aware that, in an 
important sense, Curie's statement is false. It is possible for a symmetric 
system to behave in an asymmetric fashion. This phenomenon, known 
as symmetry breaking, is an important mechanism underlying pattern 
formation in many physical systems from astronomy to zoology. The 
mathematical theory of symmetry breaking provides a powerful method 
for analyzing how symmetric systems behave and applies across the 
entire range of scientific disciplines. 12 

Curie Was Right . . . 

At first glance, Curie's statement is "obviously" true. If a planet in 
the shape of a perfect sphere acquires an ocean, that ocean will surely be 
of uniform depth, hence itself a sphere. The spherical symmetry of the 
planet is reflected in a corresponding spherical symmetry of its ocean. 



It would appear bizarre if, in the absence of any asymmetric cause, the 
ocean should decide to bulge unevenly. 

On the other hand, if the planet rotates breaking the spherical sym- 
metry and replacing it by circular symmetry about the axis of rotation 
then the ocean will bulge at the equator, preserving the circular symme- 
try. Isn't that typical of how symmetry behaves? Not always. 

Curie Was Wrong . . . 

Curie's principles may seem obvious, but they must be interpreted 
very carefully indeed, for there are many symmetric systems whose be- 
haviors are less symmetric than the full system. For example, if a perfect 
cylinder, say a tubular metal strut, is compressed by a sufficiently large 
force, it will buckle. 28 The buckling is not a consequence of lack of sym- 
metry caused by the force: even if the force is directed perfectly along 
the axis of the tube, preserving the rotational symmetry about that axis, 
the tube will still buckle. Buckled cylinders cease to be cylindrical 
that's what "buckle" means. Similarly, a computer picture of a spherical 
shell buckled by a spherically symmetric compressive force is shown in 
Figure 11: observe that the symmetry of the buckled state is circular 
rather than spherical. 

It is important to understand that the loss of symmetry in these sys- 
tems is not merely a consequence of small imperfections: asymmetric 
solutions will exist even in an idealized perfectly symmetric mathe- 
matical system. Indeed, such a "perfect" system largely controls how 
symmetries can break. However, imperfections play an important role 
in selecting exactly where. For example, when a perfect system such as 
the sphere in Figure 1 1 buckles, the axis of circular symmetry can be 

FIGURE 11. Symmetry-breaking buckling 
of a uniform spherical shell subjected to 
uniform external pressure. The shell buckles 
in a cylindrically symmetric fashion. 


any axis of the original sphere; for an imperfect system some axes will 
be preferred, their positions being related to weaknesses in the spherical 
shell. The general form of the buckled sphere, however, will be the same 
in both cases. 

In this sense Curie's principles are perhaps valid for an actual physical 
system (which is necessarily imperfect) but not for an idealized model. 
Rather than attempting to resurrect Curie's principles in this fashion, 
however, it seems preferable to understand the mechanism by which 
perfect idealized symmetric systems produce behavior with less sym- 
metry. This is called symmetry breaking. It seems to be responsible for 
many types of pattern formation in nature, and it has a very well defined 
mathematical structure that can be used to understand such processes. 

What causes the symmetry to break? The answer is that natural sys- 
tems must be stable. Curie was right in asserting that symmetric systems 
should have symmetric states, but he failed to address their stability. If 
a symmetric state becomes unstable, then the system will do something 
else and that something else cannot be symmetric. 

How does the symmetry "get lost"? We answer this question by an ex- 
ample. The catastrophe machine (Figure 12), invented for rather differ- 
ent reasons by Christopher Zeman of Warwick University in 1 969, 19 ' 20 ' 31 
shows that symmetry is not so much broken as spread around. Children 
can make one and experiment with it. 

The entire catastrophe machine has reflectional symmetry about the 
center line. If you begin to stretch the free elastic, the system obey's 
Curie's principles and stays symmetric; that is, the disk does not rotate 
(Figure 13a). But as you stretch the elastic further, the disk suddenly 
begins to turn maybe clockwise, maybe counterclockwise (Figure 13b). 
Now the state of the system loses its reflectional symmetry. The sym- 
metry has broken, and Curie's principles have failed. 

Where has the missing symmetry gone? Hold the elastic steady and 
rotate the disk to the symmetrically placed position on the other side 
(Figure 13c). You will find that it remains there. Instead of a single 
symmetric state we have two symmetrically related states. 

This is a general feature of symmetry breaking. The system can exist 
in several states, each obtainable from the others by one of the sym- 
metries of the full system. For example, the buckled spherical shell in 
Figure 1 1 breaks symmetry from spherical to circular, and the circular 
symmetry occurs about some particular axis, clearly visible in the pic- 
ture. In the "perfect" system any axis is possible, but all buckled states 
have the identical shape, and they differ only by motions of the sphere. 

Children can explore symmetry breaking with simple experiments. 
They can compress a plastic ruler to find out when and how it bends. 





FIGURE 1 2. A "catastrophe machine" can be constructed easily out of cardboard and 
rubberbands. Attach a circular disk of thick cardboard, of radius 3 centimeters, to a board 
using a drawing pin and a paper washer. Fix another drawing pin near the rim of the 
disk with its point upwards. To this pin attach two elastic bands, of about 6 centimeters 
unstretched length. Fix one to a point 1 2 centimeters from the center of the disk, and 
leave the end of the other free to move along the center line as shown, for example, by 
taping it to a pencil that you can move by hand. 

They can use a spring to hold a rod upright, with the lower end resting 
on a table, then add weights to the top and watch it sway or buckle. 
They can make a "bridge" from a flexible metal strip, put weights on 
top, and watch it collapse. 

Older students can analyze the behavior of two rigid rods joined by 
a springy hinge. These models lead naturally to more subtle questions 
relating symmetry, stability, and continuous change. How does a rolling 
body move if its center of gravity changes? How do ships capsize? The 
analysis of models of such changes brings in a great deal of important 





FIGURE 13. When the rubber band is stretched, the symmetrical position of the pin 
(a) becomes unstable. Two stable positions emerge on either side (b) and (c), but neither 
of these has the symmetry of the original configuration. In this case, as in many other 
examples in nature, instability break symmetry. 

geometry, for example, tangents and normals to a curve, centers of grav- 
ity, and even coordinate transformations. 

For a more homely example, consider the flow of water through a 
hose with circular cross section. Imagine the hose suspended vertically, 
nozzle downwards, with water flowing steadily through it. This system is 
circularly symmetric about an axis running vertically along the center of 
the hose. And indeed if the speed of the water is slow enough, the hose 
just remains in this vertical position, retaining its circular symmetry. 

However, if the faucet is turned on further, the hose will begin to 
wobble. In fact there are two distinct kinds of wobble. In one it swings 
from side to side like a pendulum. In the other it goes round and 
round, spraying water in a spiral. Similar effects are often observed 
when children wash the family car. These wobbles do not possess circu- 
lar symmetry about a vertical axis: indeed, they break it in two distinct 
ways. They also break a less obvious but very important symmetry: 
symmetry in time. The original steady flow looks exactly the same at 
all instants of time. The oscillating flows do not. The time symmetry 
is not totally lost, however: both wobbles are periodic and hence look 
exactly the same when viewed at times that are whole number multiples 
of the period. This shows how the continuous temporal symmetry of a 
steady state breaks to give the discrete symmetry of a periodic one. 

Symmetry breaking is important in biology. When a spherically sym- 
metric frog egg develops, it splits into two cells and the spherical sym- 
metry is broken. At a later stage of development (Figure 14) a spheri- 
cally symmetric mass of cells, the blastula, forms; but this first develops 
















































FIGURE 14. Creation and destruction of symmetry in the development of a frog 
embryo: spherical symmetry breaks, then is restored, then breaks again into circular and 
then bilateral symmetry. 


a circular dent (gastrulation) with only circular symmetry and then a 
neural fold, leading to mere bilateral symmetry. 

Mathematically, the development of a circular dent during gastrula- 
tion is directly analogous to the buckling of a spherical shell (Figure 
11). This demonstrates that symmetry-breaking phenomena in quite 
different physical realizations can have the same underlying mathemat- 
ical structure. The unifying role of mathematics in science, one of its 
most striking and important features, is clearly visible. 

This leads directly to our motivating question: Why does a tiger, with 
its roughly cylindrical symmetry, have stripes? Blake's immortal poem 
offers no useful clue. 

Turing's Tiger 

The theory behind the tiger's stripes goes back to Alan Turing, more 
famous as one of the father figures of modern computing. Turing knew 
that chemical changes produce the variations in coloring. The chemical 
responsible for the stripes need not be the actual coloring matter; it is 
more likely to be a precursor, formed during relatively early stages of 
the tiger's development, which later triggers a series of chemical changes 
to create the stripes. However, the biological details some of which 
remain controversial are not important here. Our aim is to illustrate 
some simple and general mathematical mechanisms for pattern forma- 
tion in a familiar context. 

Turing wrote down equations for this kind of chemical change. 29 He 
solved them numerically and then made pictures of the results. He 
used to button-hole friends and show them his pictures. On some there 
were stripes, on others irregular patches. "Don't these look just like the 
markings on cows?" Turing would ask, in some excitement. 

His calculations showed that patterns like stripes or spots can be cre- 
ated by a mechanism of instability. Imagine a flat surface (mathematical 
tiger skin) that contains a uniform distribution of some chemical. This 
would in the course of time produce a tiger of uniform color, grayish 
brown all over, more like a mountain lion. But the distribution of chem- 
icals need not remain uniform: it can change. There are two important 
types of change. Chemicals at a given place react, and the reaction 
products diffuse from one place to another. 

These two types of change compete. Reaction tries to alter the chemi- 
cal mix; diffusion tries to make it the same everywhere. The mathemat- 
ics shows that when different influences compete the result is often a 
compromise. Here the simplest compromise is that the uniform distri- 
bution of chemicals begins to form ripples. If instability occurs in only 



FIGURE 15. Competing chemical forces lead 
to instabilities. In (a), instability in one di- 
rection leads to stripes; in (b), instability in a 
second direction breaks up the stripes into spots. 

FIGURE 1 6. Spiral scroll waves in a chemical reaction created by conflicting roles of 
reaction (which changes the chemical mix) and diffusion (which restores uniformity). 

one direction, then the ripples only run one way and we see stripes. 
If a second instability sets in along a perpendicular direction, then the 
stripes themselves ripple along their length and break up into spots (Fig- 
ure 15). Competing chemical instabilities may well be the fundamental 
difference, on a mathematical level, between tigers and leopards. 

Chemical reactions that can generate periodic patterns spirals, target 
patterns can be demonstrated in any school chemistry laboratory (Fig- 
ure 16). The most famous one is the so-called Belousov-Zhabotinskii 
reaction. 21 Students can analyze these patterns to find their mathemati- 
cal structure (e.g., what sort of spiral is it?). They can also use computer 






FIGURE 17. By using simple computer packages, children can 
explore reaction-diffusion in regions of different shapes. 

FIGURE 18. Computer models of patterns on animal skins show 
realistic-looking results. They also show that long thin stripes 
usually break up into spots. 

packages to solve reaction-diffusion equations on different shapes of re- 
gions and see what kinds of patterns occur (Figure 17). 

Patterns formed in the competition between reaction and diffusion 
provide good examples of symmetry breaking. The initial uniform dis- 
tribution of chemicals has greater symmetry than do the stripes or spots 
or spirals. Symmetry breaking is a very common source of natural 


patterns. And what else is the breaking of symmetry than a change 
in pattern? 

Computer models of how pigmentation-controlling chemicals might 
diffuse through the tiger's tail produce plausible markings (Figure 18). 
Long thin stripes are less stable than short fat ones and prefer to break up 
into spots. 16 This mathematical result could help explain the common 
observation that a spotted animal can have a striped tail, but a striped 
animal cannot have a spotted tail. 


Change is a phenomenon that has a direct impact on every human be- 
ing. It affects individual lives, national economies, and the future of the 
entire planet. Until recently our understanding of change came mostly 
from the traditional tools of calculus and its more advanced relatives 
and was confined to the physical sciences, where accurate numerical 
measurements are possible. 

Initially, computers served to extend the techniques of calculus, by 
making it possible to solve more difficult equations. The term "number 
crunching" captures the style. But today's computers do more than just 
crunch numbers. In particular they can represent and manipulate data 
graphically. As a complementary development, today's mathematics is 
also about far more than just numbers. It deals in structural features, 
multidimensional spaces, transformations, shapes, forms in short, pat- 

When calculus was invented, it evolved hand in hand with geom- 
etry. Over the centuries, geometric reasoning was replaced by more 
powerful but less informative analytic techniques. The emphasis 
shifted to formulas. Now, as we penetrate areas where formulas alone 
are inadequate, the emphasis is shifting back to geometry not to the 
stilted formal reasoning often associated with the school treatment of 
geometry, but to the geometry of space and shape to the mathematics 
of the visual. 

Many basic skills are involved, often as complementary pairs, to pro- 
vide two different ways to approach the same problems: 

numerical and visual, 

algebraic and geometric, 

formal and experimental, 

abstract and concrete, 

analytic and synthetic, 

algorithmic and existential, 

conceptual and computational. 


Mathematics, the science of patterns, is itself changing. For the sake 
of our future we must harness mathematics to the patterns of change. 
And to do that we must change the way that mathematics is taught, to 
create a new generation able to perceive and manipulate new patterns. 


1. Abraham, Ralph and Shaw, Christopher D. Dynamics: The Geometry of Behavior, 
Volumes 1-4. Santa Cruz, CA: Aerial Press, 1983. 

2. Arnold, V.I. Catastrophe Theory. New York, NY: Springer- Verlag, 1984. 

3. Becker, Karl-Heinz and Dorfler, Michael. Dynamical Systems and Fractals: Com- 
puter Graphics Experiments in Pascal. Cambridge, MA: Cambridge University Press, 

4. Beltrami, Edward. Mathematics for Dynamic Modelling. Boston, MA: Academic 
Press, 1987. 

5. Crutchfield, James P.; Farmer, J. Doyne; Packard, Norman H.; Shaw, Robert S. 
"Chaos." Scientific American, (December 1986), 38-49. 

6. Curie, M.P. "Sur la symetrie dans les phenomenes physiques, symetrie d'un champ 
electrique et d'un champ magnetique." Journal de Physique, 3, Series 3, (1894) 393- 

7. Devaney, Robert L. An Introduction to Chaotic Dynamical Systems. Menlo Park, CA: 
Benj amin-Cummings, 1986. 

8. Devlin, Keith. Mathematics: The New Golden Age. London, England: Penguin, 1988. 

9. Ekeland, Ivar. Mathematics and the Unexpected. Chicago, IL: University of Chicago 
Press, 1988. 

10. Garfunkel, Solomon and Steen, Lynn A. (Eds.). For All Practical Purposes. New 
York, NY: W.H. Freeman, 1988. 

1 1. Gleick, James. Chaos: Making a New Science. New York, NY: Viking Press, 1987. 

12. Golubitsky, Martin; Stewart, Ian; Schaeffer, David G. Singularities and Groups in 
Bifurcation Theory, Volume 2. New York, NY: Springer- Verlag, 1988. 

1 3. Hargittai, Istvan and Hargittai, Magdolna. Symmetry Through the Eyes of a Chemist. 
Weinheim, FRG: VCH Publishers, 1986. 

14. Mandelbrot, Benolt. The Fractal Geometry of Nature. San Francisco, CA: W.H. 
Freeman, 1982. 

15. May, Robert M. "Mathematical aspects of the dynamics of animal populations." In 
Levin, S.A. (Ed.): Studies in Mathematical Biology. Washington, DC: Mathematical 
Association of America, 1978. 

16. Murray, James D. "How the leopard gets its spots." Scientific American, 258 (March, 
1988), 62-69. 

17. Peitgen, Heinz-Otto and Richter, Peter H. The Beauty of Fractals. New York, NY: 
Springer- Verlag, 1986. 

18. Peterson, Ivars. The Mathematical Tourist. New York, NY: Freeman, 1988. 

19. Poston, Tim and Stewart, Ian. Catastrophe Theory and Its Applications. Boston, MA: 
Pitman, 1978. 

20. Poston, Tim and Woodcock, A.E.R. "On Zeeman's catastrophe machine." Proceed- 
ings of the Cambridge Philosophical Society, 74 (1973), 217-226. 

21. Prigogine, Ilya. From Being to Becoming. San Francisco, CA: Freeman, 1980. 

22. Stewart, Ian. The Problems of Mathematics. Oxford, England: Oxford University 
Press, 1987. 


23. Stewart, Ian. "The nature of stability." Speculations in Science and Technology, 10 
(1988), 310-324. 

24. Stewart, Ian. Does God Play Dice? The Mathematics of Chaos. Oxford, England: 
Blackwell, 1989. 

25. Stewart, Ian. "Chaos: Does God Play Dice?" 1990 Yearbook of Science and the 
Future. Chicago, IL: Encyclopaedia Britannica, 1989, 54-73. 

26. Stewart, Ian. "Dicing with death in the solar system." Analog, 109 (1989), 57-73. 

27. Thompson, D'Arcy. On Growth and Form, Volumes 1 & 2. Cambridge, MA: Cam- 
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28. Thompson, J.M.T. and Stewart, H.B. Nonlinear Dynamics and Chaos. New York, 
NY: John Wiley & Sons, 1986. 

29. Turing, A.M. "The chemical basis of morphogenesis." Philosophical Transactions of 
the Royal Society of London, 237, Series B (1952), 37-72. 

30. Wisdom, J. "Chaotic behaviour in the solar system." In Berry, M.V.; Percival, I.C.; 
Weiss, N.O. (Eds.): Dynamical Chaos. London, England: The Royal Society, 1987, 

31. Zeeman, E.C. "A catastrophe machine." In Waddington, C.H. (Ed.): Towards a 
Theoretical Biology, Volume 4. Edinburgh, England: Edinburgh University Press, 


THOMAS BANCHOFF has been a professor in the mathematics de- 
partment at Brown University for 23 years. A 1960 graduate of the 
University of Notre Dame, Banchoff received his Ph.D. at the Univer- 
sity of California, Berkeley in 1964. He has held a Fulbright at the 
University of Amsterdam and served as Benjamin Peirce Instructor at 
Harvard University. Banchoff is the author of approximately fifty arti- 
cles and monographs, primarily dealing with geometry, many illustrated 
by computer graphics images. His film with Charles Strauss, "The Hy- 
percube: Projections and Slicing," has received international awards. 
Banchoff s most recent book Beyond the Third Dimension: Geometry, 
Computer Graphics, and Higher Dimensions was issued as part of the 
Scientific American Library. He was awarded a Lester Ford Award for 
exposition in mathematics and the Joseph Priestley Medal from Dick- 
inson College in 1988. 

JAMES FEY is professor of curriculum and instruction and mathemat- 
ics at the University of Maryland in College Park, where he has been 
since 1969. His bachelors and masters degrees were earned in math- 
ematics and mathematics education, respectively, from the University 
of Wisconsin; in 1968 he received a doctorate in mathematics educa- 
tion from Teachers College, Columbia University. At Maryland Fey 
teaches mathematics content and methods courses aimed primarily at 
prospective and in-service teachers of secondary mathematics. His main 


research interest is the development of innovative mathematics curric- 
ula. For the past ten years that work has focused on curricula that use 
calculators and computers as learning and problem solving tools. Fey is 
issue editor for the 1992 NCTM Yearbook on calculators. 

DAVID MOORE is professor of statistics at Purdue University, where 
he has been a member of the faculty since 1967. He received his 
A.B. from Princeton University and the Ph.D. from Cornell University. 
Moore has written many research papers and several books, has served 
on the editorial boards of leading journals, and has served as statistics 
program director for the National Science Foundation. He is a fellow of 
the American Statistical Association and of the Institute of Mathemat- 
ical Statistics and is a member of the International Statistical Institute. 
Moore has a long-standing interest in statistical education; he was, for 
example, the content developer for the Corporation for Public Broad- 
casting's telecourse "Against All Odds: Inside Statistics," and is author 
or co-author of two widely used elementary textbooks, Introduction to 
the Practice of Statistics and Statistics: Concepts and Controversies. 

MARJORIE SENECHAL received her Ph.D. in 1965 from the Illinois 
Institute of Technology. Since 1966 she has taught at Smith College, 
where she is Louise Wolff Kahn Professor of Mathematics. Her field 
of research is mathematical crystallography, a broadly interdisciplinary 
subject focusing on the classification of geometrical patterns in the plane 
and in space. Senechal is the author of Crystalline Symmetries: An In- 
formal Mathematical Introduction, a monograph on crystallography for 
physicists, mathematicians, and metallurgists. Other professional inter- 
ests include the history of science and geometry education. Together 
with the chemist George Fleck, she was an organizer of "The World's 
First Symmetry Festival" in 1973 and was co-editor of Patterns of Sym- 
metry, a volume of essays based on it; in 1984 they organized an inter- 
disciplinary geometry festival which led to the volume Shaping Space: 
A Polyhedral Approach. 

LYNN ARTHUR STEEN is professor of mathematics at St. Olaf Col- 
lege in Northfield, Minnesota. He received his Ph.D. in 1965 from 
the Massachusetts Institute of Technology, and his B.A. from Luther 
College in Iowa. Steen is the editor or author of ten books, includ- 
ing Everybody Counts, Calculus for a New Century, Mathematics Today, 
and Counterexamples in Topology. He has written numerous articles 
about mathematics, computer science, and mathematics education for 
periodicals such as Educational Leadership, Daedalus, Scientific Ameri- 
can, Science News, and Science. Steen is Co-Director of the Minnesota 


Mathematics Mobilization, Telegraphic Reviews Editor for the Amer- 
ican Mathematical Monthly, and chair of the Committee on the Un- 
dergraduate Program in Mathematics (CUPM). In previous years, he 
has served as President of the Mathematical Association of America, 
Secretary of Section A (Mathematics) of the American Association for 
the Advancement of Science, and Chairman of the Conference Board 
of the Mathematical Sciences. 

IAN STEWART was born in Folkestone, England, in 1945. He was ed- 
ucated at the University of Cambridge, obtaining a B.A. degree in math- 
ematics in 1966, and at the University of Warwick, where he gained a 
Ph.D. in 1969. He has held positions at the universities of Tubingen, 
Auckland, Connecticut, and Houston and is currently reader in math- 
ematics at Warwick. His research area is nonlinear systems and bi- 
furcation theory. He has written a large number of books, including 
Concepts of Modern Mathematics, The Problems of Mathematics, Does 
God Play Dice?, and Game, Set, and Math. He is European editor of the 
Mathematical Intelligencer, and has written articles on mathematics for 
Scientific American, New Scientist, and The Sciences. His column "Vi- 
sions Mathematiques" appears regularly in the French, German, Italian, 
Spanish, and Japanese editions of Scientific American. He has worked 
in television and makes brief but regular appearances on BBC radio. 


Abbott, Edwin Abbott, 30, 49 

Absolute value, 33 

Accelerated variation, 72 

Accumulations, 26, 27 

Algebra, 1, 3, 4, 22, 37, 39, 52, 62, 65, 66, 

70, 72, 73, 74, 82, 84, 87, 88 
Algebraic expressions, 62, 64, 74, 77, 80, 

85, 86 

Algorithms, 7, 8, 33, 36, 39, 64, 65, 77-78, 
80, 83, 84, 89 

application of, 78 

bisection, 34 

combination counting, 50 

computer-based, 78 

defined, 77 

design of, 78 

development in schools, 77 

everyday, 33-35, 36, 39 

"paper and pencil," 64 

theme in mathematics, 7 

algorithms, 78 

measurement, 89 

modeling, 89 

quantification, 65 

school mathematics, 88-91 
Archimedes, 16, 84 
Area, 12, 16, 17, 24, 67 

circle, 22, 23 

and pi, 35 

rectangle, 17 

right triangle, 16 

scalene triangle, 16 

square, 35 

Arnheim, Rudolph, 170-171 
Astrology, 62 
Astronomy, 198 
Average value, 26, 27 

B ayes' theorem, 128 

Bayesian inference, 127, 128; see also 

Classical inference, Inference 
Belousov-Zhabotinskii reaction, 213 
Bias, 131 

in data 1 14 

coefficient, 52, 53 

distributions, 122, 125 
Biological instability, 212 
Blastula cells, 210 
Boxplots, see Displaying data 

Calculators, 65, 99, 100 

as complement, 78, 79 

graphing, 75 

influence of, 63, 64 

in schools, 63 

Calculus, 4, 7, 13, 14, 19, 28, 35, 50, 184, 
215, 216 

and formulas, 196 
Cantor set, 204 
Cartesian graphs, 76 
Catastrophe machine, 208 
Causation, 111 



Cavalieri's principle, 18 
Celestial body dynamics, 198 
Center, 107, 111, 114 
Central limit theorem, 125 
Chance, 95, 97-99, 136; see also Confi- 
dence Intervals, Inference, Out- 
comes, Probability, Randomness, 
Significance tests 

calculus and, 184 
computer graphing of, 1 84 
identifying patterns in, 183 
implications, 215 
in mathematical research, 7 
mathematicians, 188 
in mathematics, 2, 184-189 
models in calculus, 184 
natural, 183 

planetary motion, 198-206 
population, 189-198 
representation of, 1 84 
in school curriculum, 2, 188 
teaching of, 187 
universal concept, 189 
Chaos theory, 143, 187 
Cheops, pyramid of, 20 
Circumference, 23, 35 
Class data, 112 
Classical inference 127; see also Bayesian 

inference, Inference 

development of skills, 146 
in schools, 147 
of shape, 147-148 

Combinations, counting, 50, 51, 52-53 
networks, 160 
properties, 143-144 
tools for patterns, 160 
Combinatorics, 122 

Comparative randomized experiments, 117 
Complete graph, 5 1 
Complex numbers, 86-87 
Computational machines, 63 
Computers and computing 
algorithms, 78 
analysis, 186-187 
animation, 32 
as complement, 78, 79 
coordinates and, 39 
geometry in schools, 175 
graphics, 2, 14, 30, 161, 167-168, 184- 


mathematics, 64 
models, 89, 214-215, 216 
new number systems, 87 

shape simulation, 177 

simulation of planets, 198 

software, 75 

statistical, 99, 100, 102 

wraparound, 38 

see also Calculators, Displaying data, 

Representation, Visualization 
Conceptual knowledge, 73, 78-79 
Conditional probability, 122-124, 128 

modeling, 123 

see also Probability 

drawing, 32 

slicing, 48 

volume of, 14, 15 

intervals, 129-131; see also Bayesian 
inference, Classical inference, 
Inference, Significance tests 

statements, 129, 130 
Configuration spaces, 41-45 
Conic sections, 48 
Connections, in mathematics, 5-7 
Conservation of energy, 203 
Context, in statistics, 96, 101 
Continuity, see Topology 
Continuous approximation to discrete 

phenomena, 194 
Contour mapping, 50 
Converging variation, 72 
Convex deltahedra, 155 
Coordinates, 32, 33, 36-45 

descriptions, 41 

dimension, 32 

geometry, in higher dimensions, 39 

graphs, 75 

Correlation coefficient, 1 1 1 
Counting games, 36 
Coxeter, H.S.M., 158 
Crick, Francis, 154, 155 
Cross-sectional slices, determination of, 47 
Crystals, 155-157 

Cubes, 11, 28, 29, 30, 31, 32, 40, 47, 53, 

counting of, 53-58 

drawing, 28 

in Froebel's kindergarten, 24 

isometric projection, 28 

learning tools, 157-158 

orthographic projection, 28, 29 

slicing, 47 

Cubic kaleidoscope, 153 
Curie, Pierre, 206, 207, 208 
Curriculum, 77, 88, 92, 95, 136, 171-180 

data, 96-97 

design, 66 


quantification skills, 62-65, 77-79, 92 

see also Teaching 
Cyclical variation, 72 
Cylinders, 11 

and discs, 22 

drawing, 32 

slicing, 47 

Data, 95, 96-97, 98, 99, 100, 101, 103, 104 

curriculum, 96-97 

definition of, 96 

experiment, 1 12 

measurement, 1 13 

numbers in context, 96 
Data analysis, 102, 103-111, 115, 119, 126 

teaching of, 112 
Data bases, 65 

Data production, 102, 103, 111-118, 135 

models, 17-18,47,49 

slicing, 46-50 
Decorated cubes, 153 
Density, 16 

chaos, 195 

phenomena, and change, 8; see also 


Diagnostic methods, 99 
Difference equation (dynamic), 193, 196, 
Differential equations, 105 
Dimension, 11, 12, 13, 25, 30, 31, 32, 33, 

Dimensional analysis, 91 

and change, 8 

configuration spaces, 44-46 

dynamic events, 44-46 

similarity, 20 
Dimension, 11, 12, 13, 25, 30, 31, 32, 33, 

36, 37, 39-44, 49, 53, 58, 62, 91 
Direct and inverse variation, 72 

as part of learning, 36-37 
Dirichlet domains, 156, 157 
Discontinuities, 200 

mathematics, 184 

system, dynamic 193 
Displaying data, 27, 28, 104-105 

boxplots, 104, 108 

for children, 105 

drawing, 164 

graphic displays, 104 

histograms, 104 

mathematical models, 104 

stemplots, 104 

see also Computers and computing, 
Representation, Visualization 


Dissection, 158-160 

Distribution patterns, 105 

DNA, 148, 154 

Double helix, 148 

Dr. Matrix, 61 

Draft lottery (1970), 132-134 

cubes, 28 

as representation of shape, 164-166 
see also Cubes, Displaying data 

Dynamical systems, 192-194 

Earth, as two-dimensional surface, 37 
Edgerton, Samuel, 168-170 
Egyptian monuments, as geometrical 

examples, 20-21 
Electron microscope, 171 
Elements, 141, 176 
Elevator geometry, 39 
Equations, 70 
Escher, M.C., 165 
Euclid, 7, 43, 64, 77, 139, 140-141, 175, 


Euler's Theorem, 161 
Evolution, of number system, 74, 81 
Executive toys, 199 
Experimentation, in place of proof, 185 
Exploratory data analysis, 41, 76, 104 
Exponential growth, 190, 191, 197 

Fermat's principle, 179 
Fields, 84 
Flatland, 30, 49 

in cube drawing, 28, 31 

see also Cubes 
Four-dimensional cubes, 30; see also 


Fourth dimension, 13, 23, 30 
Fractals, 25-26, 143, 204 
Froebel, Friedrich, 11-12, 14, 15, 17, 24, 

Fruit fly experiment, 195 
Fundamental change, 7 

Galileo, 7, 168, 169, 170, 198 

Gauss, 2, 51, 87 

Geometric gifts, Froebel's, 14, 46, 47, 58 


patterns, 139 

preparation of students, 20 

series, 25 

Geometry, 1, 2, 11, 12, 13, 14, 17, 22, 25, 
26, 32, 35, 37, 38, 39, 43, 50, 173 

analytic, 13 

and children, 1 1 

implications of new approach, 215 

plane, 12 

in schools, 174-176 

solid, 12, 13 
Globes, 161-163 
Gombrich, E.H., 143, 169 
Graphical representation, 74-75, 108, 109, 

confidence intervals, 129 

see also Computers and computing, 
Displaying data, Representation, 
Greek mathematicians, 16, 64, 74, 140-141, 

168, 175 
Grouping, 55 

exponential, 25, 190, 191, 197 

factors, 25 

linear, 190 

model, 190 

phenomena, and change, 190 
Grunbaum and Shepherd, 149 
Grunbaum, Branko, 139, 164, 165 

Harriot, Thomas, 169 

He Built a Crooked House, 30 

Heinlein, Robert, 30 

Helix, 147-148 

Hexahedra, 160 

Higher dimensional spaces, 39 

Hindu mathematicians, 74 

Hypercube, 30-31, 32, 40, 50, 53-56, 

The Hypercube: Projections and Slicing, 


Hyperion, 203 
Hypersphere, 49 
Hypothesis testing, 134 

Icosahedron, 155, 164, 165 

Image reconstruction, 166-167; see also 

Representation, Symmetry, Topol- 
ogy, Visualization, 
Independent trials, 118, 119 
Induction, see Mathematical induction, 

Natural numbers and integers, 

Principle of Finite Induction 
Inequalities, 70 
Inference, 98, 102, 103, 112, 122, 126, 

127-134; see also 

Chance, Confidence intervals, 

Significance tests 
Instruction-giving, as part of learning, 36- 


Integers, see Natural numbers and integers 
Isometric projection, for drawing cubes, 28; 

see also Cubes, Foreshortening, 

Orthographic projection 

Jupiter, 201, 205, 206 

Kaleidoscope, 5, 151 

exploration of symmetry, 5 
Kelvin, Lord, 61 
Kindergarten, 11, 17, 24, 28, 47, 58; see 

also Froebel, Geometric gifts 
Kirkwood gaps, 205 
Knots, 144, 147 
Kolrnogorov vague attractor, 185 

Lagrange points, 203 
Language of mathematics, 8 
Latitude, 37, 39 
Lattice, 155-158 

one-dimensional, 156 

two-dimensional, 156-157 

three-dimensional, 157-158 
Law of Large Numbers, 125 
Law of Motion, 198 
Law of the Iterated Logarithm, 118 
Learning and Teaching Geometry, 175 
Least squares regression, 1 1 1 
L'Engle, Madeleine, 30 
Lenses, 163-164; see also Representation 
Levels of analysis, 190-192 
Linear algebra, 62, 65; see also Algebra, 

Algebraic expressions 
Logistic curve, 198 
LOGO, 37 
Longitude, 37, 39 
Lorenz attractor, 185, 188 

Man-made patterns, 148 
Mandelbrot set, 26, 185 
Manipulatives, 14 
Mapping, 168 

of quantities, 90 

Maps, 161, 162, 163; see also Displaying 
data, Representation, Visualization 
Mars, 205, 206 

abstractions, 3 

actions, 3 

attitudes, 3 

attributes, 3 

behaviors, 3 

classification, 141 

dichotomies, 4 

induction, 82-84 

modeling for change, 196 

models, 109-111, 161-163, 178, 184, 

strands, 4 

structures, 3 

comparison with linguistics, 14 

curriculum, 62, 63, 65, 66, 77, 88, 91- 
92, 95, 96-97, 136 



fundamentals of, 3 

goals, 62, 91-92 

importance of early learning, 14 

informal arithmetic, 80 

maintaining rigor of old style, 184 

pattern and order, 1-2 

as a pipeline, 4 

public perception of, 1 

and science, 184 

statistics in schools, 95-96, 100 

variety in new approach, 1 84 
Mathematics of change, 184 

levels of description, 188-189 
Matrices, 63, 65, 87 
Maxims and minims, 72 
Mean, 96, 107, 1 10, 119, 125 

of dimensions, 14; see also Dimension 

as recurring theme in mathematics, 6; 

see also Applications, Quantification 

quantitative concepts, 91; see also 

volumes, 14, 15, 16 
Median, 107 
Meteorites, 198 
Mirror geometry, 151-153; see also 

Geometry, Plane geometry, Solid 
Mobius band, 145 
Modeling, 69; see also Conditional 


Models, 109, 161-163, 178, 184; see also 
Displaying data, Representation 


importance of technical names, 145-146 

of shapes, 145 

see also Classification 

numbers and integers, 82, 84; see also 
Principle of Finite Induction 

patterns, 148; see also Patterns 
Network problems and combinatorial 

properties, 143-144 
Newton, Sir Isaac, 7, 198 
Normal distribution, 109 

lines, 33-35 

sense, 79-80, 108 

theory, 66 

use, 67, 68-69 
Number systems, 61, 62, 81-88, 92 

algebraic and topological properties, 81 

evolution of, 74, 8 1 

future of, 88 

new, 87 


experimentation in change, 194-196 

operations, 69 

representation, 73; see also Displaying 
data, Representation, Visualization 
Numerology, 62 

Orbiting patterns, 198 
Order, in numbers, 68 
Organic geometry, 185; see also Geometry, 

Orientation of shape, 145 
Origami, 151 
Orthographic projection, 28, 29; see also 

Cubes, Drawing, Foreshortening, 

Isometric projection 
Oscar II, King of Sweden, 199, 200 
Outcomes, 97, 98, 99, 120, 125; see also 


Outliers, 100, 106, 110 
Overall trends, 72 

Paper folding, as a learning tool, 150-151 
Parallelograms, 16-17, 21 

non-square, 29 
Pascal's triangle, 53 

in change, 8, 183 

in counting, 56 

formation of, 149 

identification of, 1 

in mathematics, 8 

modeled by numbers, 61 

natural, 149 

see also Connections, Mathematical, 

Pendulums, 45 
Pentahedra, 160 
Perfect whole, 151 
Period-doubling cascade, 185 
Permutations, 55 

portraits, multidimensional change, 202- 

space, 201-202 
Pi, mathematical constant, 15, 23 

definition of, 35 

estimation of, 35-36 
Piaget, 144 
Place value, 74, 80 

evolution of, 74 

notation, 74 

numerals, 74 

Plane geometry, 12, 13, 14; see also Geometry 
Planes and surfaces, 36 
Planetary motion, 198-206 
Poincare\ Henri, 2, 199, 200-202 



Polyhedra, 144, 145, 147, 151, 154, 155, 

158, 159, 160, 161, 173, 178 
slicing of, 48 

Polynomials, 63, 81, 84, 85, 86, 87; see 
also Number systems, Principle of 
Finite Induction 
Polytopes, 158 
Popular Science, 106 
Population dynamics, 189-198 
Predicting outcomes, 97, 98, 99; see also 


Prime factorization theorem, 84 
Principle of Finite Induction, 82, 83; see 
also Natural numbers and Integers, 
Number systems, Rational numbers, 
Real numbers 
Probability, 68, 95, 98, 102, 103, 109, 110, 

118-128, 132-136 
basics, 120 

conditional, 122-124, 128 
question, 132 
runs, 120-121 
theory for children, 98-99 
Problem solving, 99 
Procedural knowledge, 73, 78-79 

of number use, 69 

of reflected shape, 151; see also Shape, 


Provided data, 1 12; see also Data 
Psychological research; see Quantification 
Pyramids, 15, 18, 19, 20, 21, 50 

slicing of, 48 

Pyrite crystal; see Decorated cubes 
Pythagorean theorem, 17, 40, 42, 43, 85 
Pythagoreans, 61 


applications, 65 

attributes, 61 

coding, 67 

data and children, 62 

everyday, 79 

fundamental concepts, 66 

information, 65 

interpretation of, 62, 79-80 

literacy, 65, 90 

measuring, 14, 67 

ordering, 67 

order of magnitude, 79 

psychological research, 66 

reasoning, 62, 67, 92 

relationships, 66, 72 

school curriculum, 62-65, 77-79, 92 

technology, 62 

of variation, 135 
Quartiles, 107 
Quartz crystals, 155 

Radio telescope, 171 

Random variables, 125, 126, 127; see also 


Randomness, 97, 98, 99, 115, 116, 117, 
120, 124-127, 128, 129-134 

in outcomes, 98, 120, 125 

in sampling, 115-117, 124-127, 129- 
134; see also Confidence intervals, 
Inference, Outcomes, Probability, 
Significance tests, Uncertainty 
Rate, 26-28, 113 

of change, 70, 72 
Ratio, 35, 203, 205 

and proportion, 20, 22, 23 
Rational numbers, 84, 85, 86 
Real numbers, 84-85, 86 
Recurring concepts, 8 
Regular polyhedra, 154-155 

discovery of, 140 

see also Polyhedra 
Relativity theory, 43, 44 
Renorrnalization groups, 143 
Rep-tiles, 159 
Repetition, and change, 8 
Representation, 62, 161-168 

computer graphics, 167-168 

drawing, 164-166 

image reconstruction, 166-167 

lenses, 163-164 

maps, 161-163 

models, 109, 161-163, 178, 184 

of numerical ideas, 73 

shadows, 163-164 

see also Displaying data, Visualization 
Resonance, 203-204 
Rossler attractor, 185 
Rubber sheet 

dynamics, 200 

geometry, 144 

Sampling, 99, 115, 116, 124, 127, 130 

distributions, 125, 129 

see also Probability, Randomness 
Saturn, 203 

Scale models, 161, 178 
Scaling, 143 
Scatterplot, 110, 132 
Science for All Americans, 72 
The Sciences, 170 
Scoring, as a learning aid, 36 
Self-congruence, see Symmetry 
Self-similarity, 142-143, 202 

in lattices, 159 

Semiregular polyhedra, 155; see also 
Polyhedra, Regular polyhedra 
Shadow and scale diagrams, 20 
Shadows, 163-164; see also Representa- 




aims of study, 140 

analysis, 148 

applied to the real world, 171-173 

classification, 140 

in geometry curriculum, 175-176, 17 

importance, 140 

interdisciplinary nature, 177 

as laboratory science, 177-179 

molecules, crystals, atoms, 148 

open-ended study, 180 

orbital patterns, 146 

and pattern, curricular issues of, 171 

patterns, 139 

and preschool children, 141 

properties, 140, 146 

protein subunits, 155 

viruses, 155 
Shear transformations, using Cavalieri's 

principle, 18; see also Volume 
Sierpinski gasket, 25 
Significance tests, 131-134 
Silicon chip, 173 
Similarity, 20 

defined, 141 

geometry, 175 

in quantification, 69 

see also Self-similarity 
Simplex, subsimplices, 53 
Simulation, 11, 102, 106, 123, 126, 129, 
130, 134 

computer, 109, 120, 125 
Skewness, 106 

Slicing, 46-50; see also Cones, Cubes, 
Polyhedra, Pyramids, 
Solid geometry, 12-13; see also Geometry, 

Plane geometry 
Spheres, 43, 49 

drawing, 32 

slicing, 47 

volume of, 14, 15 
Spiral, 147, 148 

Spread (dispersion), 107, 111, 114 
Squares, 40, 53-58 
Stable systems, 199, 208 
Standard deviation, 108, 109, 110, 125; see 

also Spread 
Star Wars, 101 
Statics and dynamics, 44 

designs, 112, 115-118 

inference, 103; see also Inference 

significance, 132; see also Inference, 
Significance tests 

uncertainty, 133 

Statistics, 95, 96, 97, 99, 100, 111, 113, 
126, 127, 134, 135, 136 

Stepped variation, 72 
Stereoscope and stereoscopic pairs, 167 
Strands, combining multiple, 7 
Subjective probability, 128; see also 

Inference, Probability 
Symbol sense, 80-81 
Symmetry 5, 6, 47, 55, 56, 142-143, 206-210 

children and, 150, 174 

in data, 108 

discovering, 150-151 

lattices, 156, 157 

packing arrangements, 155 

recurring theme in mathematics, 6 

relationships, 208 

significance and use, 153-155 

through reflection, 151-153 

see also Cubic kaleidoscope, Kaleidoscope 
Symmetry breaking, 206, 208, 212, 214, 215 
Symmetry group, 151 

Taxicab geometry, 37, 39 
Taxonomy of number use, 68; see also 

Number use 
Teaching, 187-188 

formal proofs, 187 

random events, 120 

see also Curriculum 
Technology transfer, 184 
Tesseract, 30; see also Hypercube 
Tetrahedra, 48, 16 
Tilings and patterns, 11, 149 
Time, as the fourth dimension, 43 
Titan, 203 

Topology, 144-146, 200-202 
Torus, 38-39 
Trapezoid, 29 

as incomplete triangle, 20-21 

counting, 52 

Truncated pyramid, 20 
Tukey, John, 103 
Turing's tiger stripes, 212 

Ueda attractor, 185 
Ultrasound, 171 

and change, 8 

everyday 98-99 

significance of, 135-136 

order in, 98 

variation as fundamental skill, 136 

see also Chance, Inference, Outcomes, 

Probability, Randomness 
Uniform growth, 197 

Variables and relations, 69, 70 
Variance, 125 


Variation, in process, 135 

Verhulst law, 193-196 

Vertices, 39 

Video games as part of learning, 37 


imaging of geometry, 184 

representation, and change, 8 
Visual Thinking, 170 
Visualization, 6-7, 8, 168-171 

computer-aided, 145 

dimensions, 24 

of geometric relationships, 28 

in geometry, 49 

importance of, 168 

interdisciplinary nature of, 171 

as interpretation, 170-171 

in learning, 1 1 

multidimensional data sets, 41 

problems, 47 

quantitative relationships, 74 

recurring theme in mathematics, 6 
Volume, 14-22, 26 

Cavalieri's principle, 18 


concept of, 14-16 
cone, 14, 15 

diagonal decomposition, 18 
displacement, 16 
in education, 14 

incomplete/truncated pyramid, 20-2 1 , 
irregularly shaped objects, 16 
pyramid, 15-16, 20, 21 
shear transformation 
sphere, 14, 15, 16 
Voluntary response samples, 115, 116 

Watson, James, 154, 155 
Weiner, Norbert, 1 
Wraparound, 34, 38 
A Wrinkle in Time, 30 


investigations, 155, 173 
tomography, 50 

Zeman, Christopher, 208 

Credits for 

New Approaches To Numeracy 

Executive Director, Kenneth M. Hoffman 
Staff Coordination, Linda P. Rosen 
Senior Project Assistant, Jana Godsey 
Senior Project Assistant, Carol Metcalf 


Editorial Coordination, Sally Stanfield 
Marketing Coordination, Barbara Kline 
Production Coordination, Dawn M. Eichenlaub 
Cover Design, Francesca Moghari 
Graphics, James Butler 


American Mathematical Society 


We wish to thank the Carnegie Corporation of New York for support of 
the development, publication, and dissemination of this document. We also 
wish to thank the Andrew W. Mellon Foundation for their support of further 


Texas Instruments, Inc. (p. 63) 
Stan Sherer (pp. 148, 152, 178) 
Marjorie Senechai (p. 149) 


Thomas Banchoff 
Davide Cervone 
David Moore 



J. Weeks, The Shape of Space, Marcel Dekker, Inc. (pp. 143, 144) 
P. Stevens, Patterns in Nature, Little, Brown & Company (p. 147) 

Color Treasury of Crystals, Crescent Books (p. 152) 
RJ. Hauy, Cristallographie, Cultures et Civilisations (p. 156) 
B. Grunbaum and G.S. Shephard, Tilings and Patterns, W.H. Freeman (pp. 157, 159) 
NJ.W. Thrower, Maps and Man, Prentice-Hall (p. 162) 
D. Bruyr, Geometrical Models & Demonstrations, J. Weston Walch (p. 164) 
"Draftsman Drawing A Reclining Nude," by Albrecht Durer in Complete Engravings, 

Etchings & Woodcuts, Courtesy of the Library of Congress (p. 165) 
Mathematical Association of America (p. 165) 
M.C. Escher Heirs, Cordon Art Baarn, Holland (p. 166) 
B. Grunbaum, "Geometry Strikes Again," Mathematics Magazine, Mathematical Association 

of America (p. 167) 

Thomas Harriot's lunar drawing, July 26, 1609, Petworth Museum (p. 169) 
Galileo's lunar drawing, Biblioteca Nazionale (Florence, Italy) (p. 169) 
IBM-Almaden Research Center (p. 170) 

W.A. Bentley & WJ. Humphries, Snow Crystals, Dover Publications (p. 174) 
R. Abraham and J. Marsden, Foundations of Mechanics, Addison- Wesley Publishing 

Co., Inc. (p. 185) 
R. Abraham and C.D. Shaw, Dynamics: The Geometry of Behavior, Aerial Press 

(p. 185) 

I. Stewart, Does God Play Dice? Basil Blackwell, Inc. (pp. 185, 199, 201, 205) 
J.M.T. Thompson & H.B. Stewart, Nonlinear Dynamics and Chaos, John Wiley & 

Sons Ltd. (pp. 185,203) 
G. Oster in S.A. Levin (ed.) Studies in Mathematical Biology, Mathematical Association 

of America (p. 195) 

T. Poston, Pohang Institute of Science & Technology (p. 204) 
M. Shumway, "Stages in the Normal Development of Rana Pipiens I, II," Anatomical 

Record, Alan R. Liss, Inc. (p. 211) 
J.M.T. Thompson, Instabilities and Catastrophes in Science and Engineering, John 

Wiley & Sons Ltd. (p. 211) 
A. Winfree and S. Strogatz, Mathematical Intelligencer, Volume 7, No. 2 (cover), 

Springer- Verlag Publishers (p. 213) 
J.D. Murray, "How the Leopard Gets Its Spots," Scientific American. Illustration by 

Patricia J. Wynne (p. 214) 

New Approaches to Numeracy 


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